Monday, April 21, 2008

From Leonard Susskind to Everyone:

A number of years ago I became aware of the large number of physics enthusiasts out there who have no venue to learn modern physics and cosmology. Fat advanced textbooks are not suitable to people who have no teacher to ask questions of, and the popular literature does not go deeply enough to satisfy these curious people. So I started a series of courses on modern physics at Stanford University where I am a professor of physics. The courses are specifically aimed at people who know, or once knew, a bit of algebra and calculus, but are more or less beginners.

The response was overwhelming and it was suggested that Stanford put them up on the internet. You can find them at

Since the videos went up, I have received many emails with good questions. Some are about the material in the courses. Some are more broadly about physics and science. Here is the place to ask them. If I know the answer to your question I will post it. If not perhaps someone else can answer.

Leonard Susskind


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Anonymous said...

I think its great that you've started this blog. Looking forward to seeing the Q&A.

Anonymous said...

I am currently following the Itunes QM lecture course 2006 and in PHY24 lecture 4 I could not follow the "counting" that a 2 electron system would require 6 independent real variables to specify its state, considering that the 1 electron system requires 2. Can you recommend a book or contact that could assist with this question?

Leonard Susskind said...

Dear jpr,
Your question gets to the heart of the matter. One would think that if it takes 2 real parameters to specify the state of an electron spin, then it would take 4 to specify the state of two electron spins. But as I will explain, it takes six. This means that there are states of the two electron system that are more general than the "product states" of two un-entangled electrons. The extra states are of course the entangled ones. Now lets do the counting.

Here is the rule: If the dimension of the space of states is N then it takes N complex numbers to specify the components of a general vector. That means 2N real numbers. But there is a constraint that the sum of squares is 1. Also the overall phase does not matter. Therefore the number of real parameters is 2N-2.

Now consider the spin of a single electron. The space of states has dimension 2 (up, down). Then as you say, the number of parameters is 2N-2 = 2.

Now consider 2 electrons. The space of states is 4 dimensional (uu, ud, du, dd). Now N=4 and 2N-2=6. So as I said, there are more states than you might expect.

I hope that helps.

Leonard Susskind

Clinton said...

Here's the location of scans of professor Susskind's lecture notes for the current Special Relativity and Field Theory classes:

Clinton said...

Heres' a dramatic representation of a vector field.

This display shows the SF Bay wind pattern as "streaklines".

JH said...

Dr. Susskind,
Can you recommend a reference(s) book.

Leonard Susskind said...

There is a book that I like a lot for the subject of classical field theory. "Classical Field Theory" By Davison E. Soper, 1976. I think it is published by Dover.

David Griffiths' Introduction to Electrodynamics (3rd Edition) is also excellent.

Leonard Susskind

Anonymous said...

In CM we did the harmonic oscillator. It's a great example for many reasons, but I have no intuition about Hooke's law, it's magic to me. So I've read up on gravity, the 2-body central force problem. I think I understand most of the derivation, using Noether's theorem and changing the Lagrangian to polar coords to explicitly get that angular momentum is constant, letting us use angle in place of time, after which we get a simple harmonic oscillator equation with a constant driving force. My question is, if the initial condition is both bodies at rest or with only radial velocity, the switch to theta won't go. The equation of motion is simple, but nonlinear. That is, r_tt = 1/r^2, ignoring constants, and that's not really a partial sign, this is an ODE. We can still change variables to u=1/r, but it doesn't look any better. Did I make a mistake, is it OK and easy to solve, or is it brute force from there? Thanks, Mike

Anonymous said...

OK, think I got it. Conservation of energy says T+U=E=const. Integrate F=-1/r^2 to get U=-1/r. We have T=1/2 m rdot^2. Solve T+U=E for rdot. Cross multiply to get dt on one side and nothing but r terms on the other, times dr. Integrate both sides and we have t as a definite integral of r. Thanks, Mike

Leonard Susskind said...

Excellent question. The problem is that the extra constant driving term is inversely proportional to the angular momentum L. When the orbit has no angular momentum something bad happens to the equation. Of course what is happening is that the orbit has degenerated to a straight line (infinitely narrow ellipse). The angle is no longer defined and it doesn’t make sense to ask for the radius and a function of angle. But it still makes perfect sense to ask for r as a function of time as long as the orbit stays away from the singularity at r=0 where the force becomes infinite.

Unknown said...

With Ref to PHY 25 Classical Mechanics: Early on, you defined pi(i) as the "momentum canonically conjugate to q(i)". The definition I found for "canonically conjugate variables" was " variables always occurring in complementary pairs". This seems to fit your usage. Later on, in lecture 9, you use "canonical" in describing "canonical transformations".

Limiting the search to physics and mathematics, I have looked up "canonical" in every source I could find, and found a variety of meanings:

1) Conforming to well-established rules or patterns
2) Reduced to the simplest and most significant form possible w/o loss of generality
3) The concept of uniqueness or naturalness
4) Simplest or standard form of an equation, coordinate, etc.

I am very confused about exactly what "canonical" means, and would appreciate it if you would explain your use of the word.

Leonard Susskind said...

fheinzmann asks about the meaning of canonical in classical mechanics. It is of course a technical term with a very specific meaning. Every coordinate of a mechanical system has a corresponding momentum. Thus the variables come in pairs (q_i and p_i)
where q = coordinate and p=momentum. Each p_i is obtained by a standard (canonical) procedure: differentiate the Lagrangian with respect to the velocity, p_i = dq_i/dt. The term canonical means standard.

The space of p's and q's is called phase space. Canonical transformations are the special transformations of phase space that preserve the mathematical connections between coordinates the each pair (q_i,p_i). Technically, canonical transformations are the transformations that preserve the Poisson brackets between the coordinates and momenta.

Leonard Susskind

Unknown said...

Dear All,

My query is from Cosmology (& Particle Physics). Here is a brief intro:

The early universe was radiation dominated. As you look earlier and earlier, things get hotter. When kT>mc^2 particles of mass m behave as radiation (rather than cold matter), and contribute to the number of relativistic degrees of freedom. [There are some subtleties if you want to do this accurately, but this is the basic idea].

The number of degrees of freedom is thus higher at earlier times. At temperatures ~ TeV all of the particles in the standard model contribute to a total of just over 100 degrees of freedom.

At yet higher temperatures, heavier particles may be expected to contribute.

My question is:
What particles are predicted at the GUT temperature and PLANCK temperature by supersymmetry and string (or M) theory ??? How many degrees of freedom do these contribute ???

The number of degrees of freedom at the Planck time may be of particular interest because it determines the current temperature of the background of gravitons left over from the big bang. .... Their detection would revolutionize cosmology in the way the CMB observations have in the past decade.


Chas Egan,
Australian National University

Leonard Susskind said...

Chas asks how many particle types are there at each energy (or temperature) scale. Roughly speaking, if we heat space to temperature T then each particle species (count spin states and antiparticles as separate species) with mass less than T gets produced by the thermal fluctuations, and moreover, each species gives a thermal energy proportional to T^4, but only if its mass < T. The total energy is therefore T^4 time N(T) where N(T) is the number of particle species with mass < T. The function N(T) is fundamental to the energy density in the early hot universe and determines how rapidly the universe expands in its early stages.

We basically know everything about the particle spectrum up to masses of about 150 times the proton mass and that tells us the energetics up to the corresponding temperature. But it does not take us back to the earliest times.

So what do our “best” theories tell us about elementary particles with mass > 150 times the proton? The hypothetical supersymmetry invoked by theorists roughly doubles the number of species above that mass. Each ordinary particle has a partner that only reveals itself at higher mass. If I count correctly the, the number of particles in the usual standard model is about 35 and supersymmetry would double it. Of course no one knows if supersymmetry exists but if it does then the count would be 70.

But no one believes the count ends there. An tremendous masses of about 10^{16} proton masses, theorists suspect the existence of a more unified theory that would have another dozen or more particles. If the universe was this hot these particles would create pressure and cause it to expand faster than other wise.

Above that mass things become murky. String theory suggests that there are many more particles and above the Planck scale there are black holes which are the natural extension of the particle spectrum. But it is pretty clear that the rules break down. Instead of the number of independent degrees of freedom increasing as the mass increases, the opposite should happen—the number of degrees per unit volume must decrease! This is called the Holographic Principle. But no one knows if the universe was ever hot enough for this to have been important. I have my prejudices but that’s all they are: prejudices.

As Chas points out, the physics of very high temperature would determine how many primordial gravitons are floating around in analogy with the cosmic microwave photons. The gravitons could only be produced by Planck scale energies (temperature). But there is no way now to know if the universe was ever that hot. String theory suggests a limiting temperature which could be a good deal lower, in which case gravitons might be less abundant. But we are way out of out depth at the moment.

So after all that, all I can say is that I don't know.

Leonard Susskind

Leonard Susskind said...

One other point I should have added to my answer to Chas: The inflationary theory of cosmology would have diluted the primordial gravitons as it stretched the universe. The temperatures associated with inflation are way below the Planck scale, so there is no mechanism for significantly replenishing them. In other words, inflation puts a practical limit on the early energy density of the universe which is likely to be many orders of magnitude lower than the Planck scale.

Anonymous said...

My take on "canonical" worth .02 cents. Yes, "canonical transformation" seems like sense (4), "standard". To me, "canonical momentum" seems like sense (1), "rule based". I've seen variants "canonical momentum conjugate to q_i", "generalized momentum conjugate to q_i", and "i'th conjugate momentum". The common theme is it's never just "momentum". By itself, the reader may assume we're talking about ordinary, cartesian, linear momentum. But if we're in polar coords with q_2 = theta, then what we're talking about would be angular momentum. So, not literally "momentum", but metaphorically. Generalized coords can be any degree of freedom of a system, so the metaphor could be a stretch. By rule, we'll call it "canonical" momentum. As for "conjugate to q_i", we might say "in the i'th direction" but we could have many q_i's in the x direction. We might say "along the i'th degree of freedom". Calling this relationship "conjugate" emphasizes that, as Prof. Susskind answered, q_i and p_i complement one another. -Mike

Anonymous said...

I meant to say we call Pi_i the "canonical momentum" because we get it via a "momentum rule", stated by Prof. Susskind:

Each p_i is obtained by a standard (canonical) procedure: differentiate the Lagrangian with respect to the velocity, p_i = dq_i/dt.

On a related topic, I found helpful the following more even-handed description of the Legendre transform. I'll gloss over partial derivatives and write ordinary derivatives until I can't stand it.

Suppose we start with either f(u) OR g(v) and we want to make its partner. That is, if we have f(u) we'll construct both v and g(v), or VICE VERSA. Their relationship is:

f(u) + g(v) = uv

Now we take derivatives.

(df/du) du + (dg/dv) dv = v du + u dv


v = df/du
u = dg/dv

For all other coords, besides u and v, we define f and g to agree: f(x) = g(x), f(t) = g(t), etc. That means

partial_x (f + g) = partial_x (uv) = 0
partial_t (f + g) = partial_t (uv) = 0

This means

partial_x f(u) = -partial_x g(v)
partial_t f(u) = -partial_t g(v)

Now let f=H, g=L, u=p, v=qdot. Then

H + L = p qdot
p = dL/d(qdot)
qdot = dH/dp


partial_t H = -partial_t L
partial_q_j H = -partial_q_j L

for all q_j =/= q. All this just from the math. No physics yet! Now we bring in the equations of motion, i.e., Euler-Lagrange.

-dH/dq = dL/dq = (d/dt) dL/d(qdot) = dp/dt = pdot

Thus we get the second Hamiltonian equation. We already got the qdot equation in the Legendre recipe.

(d/dp, d/dq and d/d(qdot) in these equations are all partials.) One more equation falls out of the dynamics.

dH/dt = partial_t H

This is because {H,H} = 0.


Anonymous said...

Oh, correction. Actually, v=qdot, so

partial_q H = -partial_q L

for all q. Sorry about that. -Mike

Anonymous said...

I'll add one thing I forgot. The Legendre recipe is repeated over all qdot_i, each time starting with the H, say H_i, from the preceding step. The final H_n = the full H.

I'll also paraphrase this equation:

Hdot = dH/dt = partial_t H + {H,H}
= partial_t H

In general

Adot = partial_t A + {A,H}


Anonymous said...

...and at the n-th iteration, we'll have

H_n + L = sum/i p_i qdot_i
= p dot qdot (vector notation)

p_n = del L/del qdot
qdot = del H/del p

Prof larnt us last night that a dot on top may not always mean total time derivative. It's often convenient to let a dot mean partial_t at the cost of needing d/dt for total time derivative, which may however be less needed. Someone suggested in email "del" = "partial" and for that upside-down triangle thingie, call it "grad". So I'm trying that.

dA/dt = del A/del t + {A,H}
dH/dt = del H/del t + {H,H}

= 0, if energy is conserved. :-)

Prof. Susskind,

I'd like to point out another resource. Of course we have a Google group only for students physically attending class, a good spot for those logistics. That group is by invite only. I can infer from emails I get that some students never activate. We also have the old student-run Yahoo group. Many current and former class members are on this list and still get the emails. I believe you're on it, too. It may be a good idea to copy announcements of general interest--i.e., not class specific--to the Yahoo group.

I imagine study-group discussions might be appropriate there. People with ideas and tips could air them out. Questions needing an expert answer could be directed to your blog. What do you think?

If this makes sense, I can ask for feedback. I don't want a backlash over too much email. It could be significant, as I'd like to invite readers of your blog to join the student-run Yahoo group. If it passes muster, you might want to unsubscribe, inasmuch as you wanted the blog to lighten your email.


Unknown said...

Last night in Lecture 5 of the Special Relativity course, we used a complex partial deritative to find a conserved quantity with Noether's theorem.

I found an explaination of the mathematics at:

in section 3.

But could you explain why the conserved quantity uses the two terms:

Int( dx [Π_φ* f_φ* +
Π_φ f_φ])


Clinton said...

The 5/12 Fields class uses the wave equation Lagrangian to demonstrate the E=cP relation between energy and momentum. Is there a way to demonstrate that relation using a general Lagrangian, rather than a specific example?

Clinton said...

May 12 lecture notes are up!

Anonymous said...

Readers of this blog may be interested in a student-run Yahoo group called "Stanford Quantum Groupies". That name has been posted on the board in class a few times and is probably in the videos. Go to and type our name into the search box. You can read without joining. To post, you need to join. Mumble something about being interested in physics. Cheers, Mike

Anonymous said...

In last night's lecture, Prof Susskind did some SR. He didn't waste Greek letters on certain customary values, instead just writing whatever came up, like dt/d(tau)=1/sqrt(1-v^2). (He let c=1 temporarily.) Someone asked him about the Greek letters. He said they're fine, but it's one more thing to remember, and forget, and re-remember. What this made me think of was the hyperbolic trig he did awhile ago.

We have beta=v/c =tanh omega (=v when c=1) and the "Lorentz factor" gamma=1/sqrt(1-v^2/c^2) =cosh omega. Dropping omega, beta=tanh, gamma=cosh. Our main identity is cosh^2 - sinh^2 = 1. Dividing by cosh^2, 1 - tanh^2 = 1/cosh^2. For v=0, we get tanh=0, cosh=1. For v near c, we get tanh near 1, cosh=big. The Greek letters in the usual Lorentz transformation are gamma & (beta gamma), which are of course just cosh & sinh. -Mike

Clinton said...

The February 18 QM note page now has a pointer to additional student notes:

Notes are also published for the May 19 class.

Clinton said...

How to set up quantum mechanical operators in curvilinear coordinates (the basic energy and momentum operators, with their conjugate coordinates)?

Using Lie derivatives as 4-momentum operators appear to be the way to explore given their connection to conservation laws and with their easy identification in a Minkowski metric.

The going gets tougher if you demand covariant conjugate coordinates with respect to the operators in curvilinear coordinates.

The goal is to find covariant operators, probably Lie derivatives, that mimic the familiar operators.

kmatusow said...

Quick question, does the vacuum energy have any refractive properties? The refractive index n=c/v(p) where v(p) represents the phase velocity of light in a particular medium. Can the vacuum energy be considered a 'medium'? If so, what is its refractive index?

JeffJ said...

I've read in several QFT books that the quick way to go from classical mechanics to quantum mechanics is to convert the q's and p's to operators and impose a communtation relationship between q's and p's. I don't understand why this quantizes the mechanics. What have I missed?

Anonymous said...

Hi Jeff. Quick is relative, I guess. You mentioned the commutation relationship.

{p,q} = 1 --> [P,Q] = -i hbar

That's only the first step. Converting p and q to operators doesn't mean put a hat over them. Write them out.

qhat = multiply by q
phat = -i hbar d/dq
Ehat = i hbar d/dt

Classical KE p^2/2m converts to -hbar^2/2m del^2. (The Laplacian.) Omitting PE,

i hbar d/dt = -hbar^2/2m del^2

(Schroedinger.) As opposed to the classical equations of motion d/dt = {,H}. (The PB with H.)

Does this help? -Mike

JeffJ said...

I'm still confused. I understand how it works in the specific case of the Schroedinger equation and quantum mechanics. I don't understand why it works in general. Which comes first, the commutation relation or the form of the operators for q and p? Doesn't the form of the operators determine the commutator? What does it mean to "impose" a commutation relationship?

Anonymous said...

Prof. Susskind emphasizes that CM comes from QM. So, not why is QM quantized, but why isn't CM? Quantization works backwards, like crime scene evidence. What connects CM to QM? What are the steps in "quantization"? What good is the commutator? If it's "fundamental", why is it seldom used, besides the uncertainty derivation? The "forwards" direction from QM to CM is Ehrenfest:

i hbar d/dt < Q > = <[Q,H]>

The reason CM isn't quantized is because of averaging.

The commutation relations come first in the sense that the above form of Q and P are in position space. In momentum space, we'd get Q = i hbar d/dp. There are other "bases" (CSCO's). Generalized coords and canonical momenta might not be business as usual. The form of our operators will reflect both that and our choice of basis (representation). What seems to be key in the quantization procedure is

[U,V] = i hbar {u,v}

PB's between p's and q's are always 0 except between conjugates, the q and the p of the same index (degree of freedom): {q_i,p_i} = 1, [Q_i,P_i] = i hbar. This is "imposed" in that P's and Q's that break the rules are out the window. Learning CM, I often got confused because I didn't pay attention to the roles of things.

H drives the p's and q's via the equations of motion. The p's and q's aren't just coords, but are our solutions to those equations. If a "dynamical variable" depends on the p's and q's, we can of course solve and plug them in, or we can bypass them and let H drive that variable via custom equations of motion. The shorthand d/dt = {,H} says that all the equations of motion of the p's and q's are built into this one equation, like how we use vector equations to combine all the equations into one. We can get back the original equations through the magic of "grouping like terms". We can also just write them down from dq/dt = {q,H} = dH/dp, dp/dt = {p,H} = -dH/dq. Take f, whose equation is df/dt = {f,H}. Then {f,q} = -df/dp, {f,p} = df/dq. In this way, we can get all the terms of f's equations of motion by calculating PBs. When we quantize, there may be some basis where the wave equation is especially simple or easy to solve.

Basically, we may be able to do some of our math up front using PBs or commutators without minding our P's and Q's. So I gather. There's a short chapter in Shankar on The Classical Limit. On PBs, see Dirac p84, Ch4 The Quantum Conditions, S21 PBs.

Anonymous said...

delighted to see an 'outreach' blog like this by a true expert (I have been trying something similar at a much lower level).
I suspect there is great interest in such material, not just from the general public, but from other physicists in different areas
Cormac O'Raifeartaigh

Paul Sheldon said...

In lecture 7 relativity starting April 2008, someone asked a question overgeneralizing what Prof. asserted. He inquired about replacing covariant derivative for ordinary derivative in Lagrangian for constructong gauge invariant Lagrangian into the prescription for getting field equations using Euler Lagrange equations with D rather than normal derivatives.

Student rapidly backed off thinking exploring the sense or nonsense of this was going to cost too much time, but lecture 8 makes at least the question inexpensive to say.

Replace D for ordinary derivative in Euler Lagrange and would the field equations be nonsense? We already know ordinary derivatives in EL get field equations from gauge invariant Lagrangian, so a gadfly might say why bother seeing what the D modified EL eqs beget. Yet, they are not manifestly covariant and might be condemned to not be a law of nature which are.

Will the course in general relativity further connect this covariant derivative with the one I know in GR that you are supposed to express laws of nature in?

Will there be a GR EL equation?


I hope; I think I would understand more if a better mind than mine could only stretch EL there.

This is just one of those things you get scared asking google.


And I sure don't have the nerve to ask myself.

Lenny said...

At the last meeting of Susskind's class, he said, if I heard right, that the last Lagrangian he displayed can be used to prove two of Maxwell's eauations (the other two follow by definition) and can also be used to estabish the force law F = e(E + v x B). Does anyone know how to do this?

Thomas said...

Hello Professor Susskind:
I read your recent book "The Black Hole War". Thank-you for taking the time to write it. I found it to be not only a clear narration of the issues surrounding an exciting scientific debate, but also a thoughtful and respectful description of the people involved in working through the challenges posed by Hawking's question.

I have a question that occured to me as I read the book.

The Planck-Einstein equation E=hv relates frequency to energy. How does an individual photon have a frequency. In its rest frame, it has a proper time that does not change since it travels with the speed of light. I remember a long time ago I read Feynman's book QED and (if I rememebr correctly) he talked about different photon frequencies having different rates in their phase angular velocities that related to their energy.
Thank-you for taking the time to start a blog.

chumbly said...

I am looking for your lecture on the basics of Langragrian theory, can you say which lecture you started speaking about it?
ps I love your lectures, you must be God himself.

Paul Sheldon said...


Also reading Black Hole War. Information loss relates to time asymmetry which evidently has some say in whether a black hole can be a hole.

Believe some Euler Lagrange equations derive EM field dynamics while others charge dynamics.

I'm a bit afraid I'd find myself dealing with point particles and Dirac delta function fields. I must use my software to jump through last lesson and find if he left the action principle derived Lorentz force law as an assignment or claimed we weren't ready for it.

chumbly said...

In the entanglement equation for say, one state, is there a euclidean variable in this equation or an assumption about convergence? If not then the equation is valid over large euclidean particle separations. Seems unlikely that a 'complex wave' could be so spread out?

Paul Sheldon said...

Whoops I had started to answer Lenny.

Tried fast scanning last lecture, but did not find Euler Lagrange getting Lorentz force law from same Lagrangian some of Maxwell's equations came from.

I only saw Professor Susskind making another Langrangian do this. At one point he said suppose we were given j and saying dealing with it varying too complicated in the time for course, ie. course was going to cover field response to charge not vice versa.

Paul Sheldon said...


No proper time passes for photon to have frequency in its rest frame. So, were we to see a finite frequency, it would see infinite frequency.

Strangely, photon has zero rest mass, so I would expect zero not infinite frequency.

Thence, I merely amplify your question or statement of contradiction. Hope the contradiction is fruitful. Einstein's vision was that there was no rest frame for the photon, so this might be a reducto ad absurdum proving that vision...

Paul Sheldon said...


A further amplification of your question.

Photons don't "have" the frequency of the Planck formula, rather rest frames looking at photons do.

Now, try a thought experiment. Have your observing rest frame approach velocity of photon. Observe frequency Doppler shift down to zero proportional to rest mass 0!


Einstein and Planck, not in contradiction as before, are now working together!

Now, there is consistency : photon doesn't have frequency as a moving clock has frequency in its measurement of proper time. The frequency in Planck's formula is a property "had" by the observer not the observed.

Droog said...

Dr. Susskind.
Duh, posted this reposnse in the wrong post originally. What can I say..
I am one of the "curious people" that you mention in yopur original post on this topic. Please forgive my apparent rambling to begin with, I DO have a point.
I have always been a science/technology freak, something of a polymath, and was fortunate to have been born at the start of the space age, which is what first interested me (are you kidding, the moon shot was flat out cool).
Anyway, I have also always loved music, and as time passed, I got bored with most current music. (hang in, I'm going somewhere with all this..)
I have also always been a voracious reader, and over the last couple of years, I discovered spoken books, which were great replacements for music while I was spending a lot of time in my car. During my search for downloadable literature, one day I came across some particle physics courses "for beginners" and another set on cosmology. I listened to them both many times over,and began to pick up the "lingo" that made other, more detailed talks accessable to me. Then I discovered the KITP web site that has tons of talks and I have not looked back since. I was thrilled to find that there are many other sites out there with similar talks. My original goal was to make it all the way through Richard Feynman's Caltech Lectures, but the more I discover, the more I can't wait for the LHC to start colliding.
ANYWAY. My point is this, and I have been mulling this over for a while, and your blog was the first time I had seen it touched upon.
There are many people out there just like me, who burn physics MP3 discs to play in their cars, spend all their time looking for new talks (good one from you at the Perimeter Institute BTW) and are soaking up this stuff like a sponge. It's the "music" of the mind. Some of you guys are the rock stars, and some are the new artists (I really like Ayana Holloway Arce {UC Berkeley}, she gave an excellent talk at KITP on the LHC, I loved her enthusiasm).
So you are right, and kudos to you for seeing the trend.
I'm sorry that I can't do the math, but hey, for now, I'm mostly getting audio, wait until we get some decent streaming video going, I'm sure I'll pick up as much as I need to...
Say Hi to Steven for me, his book was also part of my evolution, and I'm glad that you saved the world by winning the war.
Colin Bembridge N.F.D.A.A.
Toronto, Canada.

Clinton said...

Has anyone explored whether the covariant derivative can be substituted into the Lagrangian equations of motion? For consistency the outer derivative is also a covariant derivative. Calculating the field equations would be easier if covariant derivatives could be used throughout.

Ty said...

I guess there are two threads here so I'll post (with some corrections) in this one also...

Like all the others have said, thanks so much for holding these classes and making them available on the internet. During my freshman year in college I changed my major from physics to mechanical engineering and have somewhat regretted that decision ever since. My new year’s resolution this year (30 years after my freshman year) is to learn, and relearn, as much physics as possible. These classes have been a great way to start down that road. I’m 2,500 miles from Stanford so I view all the classes on-line.

While I’m able to follow and understand all the material as presented in the lectures, to really learn the concepts I (like most folks I suppose) need to work some problems. Can anyone recommend a source for problems on the internet or in some books? I’m currently working my way through the special relativity April – June, 2008 lectures.

Many thanks.

Moonbat said...
This comment has been removed by the author.
Moonbat said...

Dear professor Susskind,

In the your lecture series Modern Physics, The Theoretical Minimum, Classical Mechanics, PHY 25 in your first lecture you gave an example of a coin-like system with two states: heads and tails. One possible law describing the evolution of that system was for heads to go to heads in the next time step and for tails to go to tails in the next time step. If that law were true why would we call that a single two-state system instead of two completely independent one-state systems? Is it just a matter of convenience (and hence if we find a system which obeys a conservation law does that imply because we are grouping independent kinds of things together) or is there some unspoken criterea involved?


chumbly said...

In the entaglement equation there is nothing to stop an state changing as far as I can see. So if an entangled electron went through a spin e field would that break the boundary conditions of that entanglement equation?

I bet you wish you had not promised to answer questions now!!

PhilG said...

The poor definition of temperature mentioned in "The Black Hole War" (pg. 168) is the same one I got in a thermodynamics course - "it's what you measure with a thermometer".
On page 169 Professor Susskind provides a "correct" definition: "Temperature is the increase in the energy of a system when you add one bit of entropy."

While I have absolutely no doubt this definition is correct, for me it does not answer his original question and now mine: What is temperature?

In relation to the stated definition, I have these questions:

Why is temperature defined using a change in the system? Why is it not defined for the system with whatever energy and entropy it has before the bit of entropy is added?

Where does the increase in energy come from when you add entropy?

What determines the amount of energy corresponding to the added bit of entropy?

Unknown said...

First off, a thousand thanks for these lectures! They have helped me enormously!

In your second Special Relativity and Field Theory lecture here, you wrote a term for the potential energy of mass i as

U = (K/2)(Phi(i) - Phi(i-1))^2

This only takes into account the effect of the little spring to one side of the mass under consideration. What about the effect of the little spring on the other side?

Shouldn't U for mass i be affected by both sides? i.e. Shouldn't it be:

U = (K/2){
(Phi(i+1) - Phi(i))^2
(Phi(i-1) - Phi(i))^2


Thanks again,


Unknown said...

question about black holes:

If Eisteins field equations are time symmetric then why is it the case that while going forward in time, black holes only allow objects to travel inward and going backwards in time objects can only travel outwards. Should it not be independent of the direction of time since the equations are time symmetric?

That is if the equations can't tell the difference between directions in time then why can we?

steve said...

Prof. Susskind,

In your book "The Black Hole War" one of the formulas that you risked including in the book was the wonderful formula for the temperature of a black hole. Using that formula to solve for mass, I set the temperature to that of the cosmic background radiation, 2.725°K, and that yielded a mass value of 4.62139x10^46kg. Would this represent a valid upper limit for the mass of the universe?

Clinton said...

Here's the location of scans of professor Susskind's lecture notes and some student notes for the Spring 2008 Special Relativity and Field Theory lectures:

and for the Fall 2007 Classical Mechanics lectures:

and for the January 2008 Quantum Mechanics class:

hyojin said...

Prof. susskinds,
i think your video lecture is great, however in the first lecture of SR lecture, you worte sin^2 h for sinh^2 omega. ^^ it is just comment.
my real curiousness is why lagrangian is always least action, why least mass or vibration for string theory?

Anonymous said...

Hello Mr. Susskind,

I am a high school physics teacher in Brasil (Londrina - Parana) and I pretend to introduce modern physics in my class (18 hours course). My question is: What is most important to teach? How can I teach Modern Physics only using high school math?

Paulo Angelico

scoop said...

From a certain point of reference, A is stationary and B is moving very fast, and so B's clock is running more slowly than A's. From a different point of reference, B is stationary and A is moving very fast, and so A's clock is running more slowly than B's. What am I missing?

Unknown said...

Hi, professor Susskind! I'm a begineer at Special Relativity(20 yrs old) but I'm so dearly interested in physics right now is that I've started decorating my room with the scientists-posters. I was always fascinated with physics and I sure did blow my whole class with my highest score ,once, evenafter I jumped classes from 6 to 8 (this was the third time that I had jumped classes). I'm not flattering myself but instead I am showing you my capability and what I can do just like others. I just want you to provide me the names of the book that would help me understand ,vividly, about Special Relativity(I am understanding a lot more from your lectures) and Quantum Mechanics! I want to get deep into physics! I do not want to do anything but just be with the physics-books. I want to understand more about "ElectroMagnetGravity" Could you help me?

Clinton said...

Those who attend Professor Susskind's GR class can find his notes at:

kmatusow said...

can someone point me to the videos of the GR class?

Ty said...

I download and view the lectures from Itunes. Go to Itunes U then to Stanford.

johnl said...

I normally download the lectures from Itunes also. However, I just checked and this fall's lectures are not there yet. Does anybody know when they will start being posted?

Ty said...

In the past they have posted the lectures in a group sometime after the last lecture takes place.

Clinton said...

Here's the location of scans of professor Susskind's lecture notes and student notes for the current General Relativity classes:

Nick said...

Dear Professor Susskind, I just wanted to say again I really enjoyed your lecture at the Ottawa Writer's Festival. After the lecture I had posed a question to you about the concept of the universe "recycling" itself through expansion and contraction and expansion again, but was unable to recall the source of the idea. I've since found the article again at the following newspaper and I was wondering if you could comment? Thanks.

mogi said...
This comment has been removed by the author.
mogi said...

Dear Professor

Is it possible that nature of space-time where time is subtructed from spatial distance while counting the space-time interval between events is strongly corelated to the special role of quantum observation? Meaning that: because speed of any signal is limited the observation always concerns an event that is in the past and as observation makes the observed object choose a particular state we may conclude that these two events (observation and observed event) are somehow in one space-time place or ar parts of one name it super-event thus their space-time distance (interval) would have to be 0.
Could it be the effect of discrete space-time in which there exist different size collections of space points where the bigger collection corresponds to fututure moment in time and its points are derivatives of points from the less numerous collection (previous moment in time). In that case future would be the past described by more parameters (bigger number of points).

Clinton said...

Please find General Relativity student class notes and scans of Prof. Susskind's lecture notes on

Clinton Lewis

Unknown said...

Iam firm believer that everything in this universe is made of waves. (i.e. fluctuation of vibrating energy). And mass/charge are just a way in which this fluctuation interacts with its surrounding.

And iam having have a question related to youngs double slit experiment.
From that experiment we conclude that if we try to observe that from which slit the electron goes, then the interference pattern is destroyed.
But has anyone tried the case as mentioned below.
Have an electron near one of the slit positioned, as if its sticked near one of the slit opening. Then cover this slit, so that when we fire the electrons it passes through the other slit only.

And then see whether we still see an interference pattern.

SwaStassijns said...

I have read Leonard's book 'The Black Hole War' with great interest.
Something which seems to be at the core of the Black Hole War problem is apparently the Equivalence principle which leads to the assumption that someone who falls through the horizon doesn't feel anything special there (no stretched horizon, no high temperatures, etc...).

I have always felt a slight itch when thinking about the equivalence principle: yes, it's very difficult for an accelerated observer to make out if he's merely accelerating or if he's in a gravity field and this way of thinking has led to the very valuable and proven theory of general relativity...but I have always thaught that there IS actually a difference! An accelerated observer will find a zero curvature tensor while an observer in a gravity field will find a non-zero curvature tensor: so the equivalence principle should actually be called 'almost equivalence principle' or something like that, no??
Which makes me wonder to which extend this (almost) equivalence principle is still valid at the horizon of a black hole...

Sergio said...

BLACK HOLE WAR - question about Alice
Quantum jitters become thermal fluctuations near the horizon as explained on page 359. These fluctuations are responsible for Bob's observation of the smashing of the atom falling together with Alice. But what happens to Alice from Bob's perspective? She is also made of atoms and therefore, every atom in her body will be also smashed from Bob's perspective. That means she will be already dead before reaching the horizon. But from Alice's perspective, nothing extraordinary happened as she crosses the horizon. She will die later when the tidal forces near the singularity become too strong (I'm assuming a huge black hole, where tidal forces are unbearable only very near the singularity).
How to explain this apparent paradox of Alice being dead and alive depending on two different perspectives?

Moonbat said...

Dear Professor Susskind i am currently working my way through the ph25 course on classical mechanics and I was wondering how much these lecture courses differed in terms of content to undergraduate lecture courses on the same topics.

Many thanks for the lectures.

Seer Magnus Prime said...

Hi Doc! I really enjoyed Black Hole War, and throughout the book, you relate, that the Zero Point Energy field is so very subtle. Does this pretty much invalidate the claims of inventors of Over-Unity devices, and ZPE energy generators? In other words, if the energy available at absolute zero is so miniscule, would not there be ample energy available @STP for energy generation if some method could be applied to access this frequency/amplitude variant? Dave

Unknown said...

Since some theories support the notion of time travel especially into the past. will time travel not violate entropy, the fact that moving a concentrated amount matter and energy would reduce the entropy (although extremely small). since time moves forward 1. the arrow of time (the big bang), 2. entropy only increases increase. 3, our ability to remember the past only.
your lectures are great, please continue I belong to the over sixty fours

Unknown said...

Prof. Sussking,
I'm new to special relativity.. however, i have a question about space-time and how to measure the path of a particle moving through space-time: In your first lecture (that i folowed on youtube) you said that the path of a particle in space-time can be measured using a simple clock delta_tau=sqrt(delta_t^2-delta_x^2)! I can't understand this since a clock measures time but her we have two axes, one is time and the other space.

Prashant Gokaraju said...

When are the general relativity lectures going to be uploaded to itunes!!?? I've been checking it constantly for weeks!

mccomplete said...

Hi Leonard. I just read (actually listened to) your book, "The Black Hole War". It was a lot of fun. I really appreciate you maintaining this blog to answer our questions. My question is about gravitational time dilation.

Let's say I am on the surface of a dense object with a strong gravitational field, such that there is time dilation to, say, a factor of 3, such that clocks far away in outer space should be running about 3x as fast as my clock.

Let's say that someone far away in outer space has a light clock, like the clocks imagined by Einstein for special relativity, where time is measured by a photon bouncing between two mirrors. Let's say that the mirrors are one light-second apart. (I realize that that is a very long distance, but of course, this is just a thought experiment.)

An observer in outer space near the clock sees the photon travel from one mirror to the other in 1 second. However, I, on the surface of the dense object, see all far-away clocks ticking 3x as fast as my clock. That means that, relative to me, the photon travels from one mirror to the other in 1/3 seconds. Which means that the photon is going 3x the speed of light.

How can this be?

sashi said...

Dear Dr Susskind,
The i's that appear in the momentum operator and the hamiltonian operator, are they fundamental to quantum theory or is it some sort of a convention? Does it have some physics attached to it?

Ty said...

Two questions:
1) Does anyone know why Prof. Susskind gave up on this blog? He has not participated since spring 2008.
2) Did they tape the General Relativity lectures last fall and is there a plan to post them to itunes?


SwaStassijns said...

with all due respect...I can imagine why Prof. Susskind gave up on this blog: some (most?) of the questions that are posted here are just too basic and other posts relate to far fetched wild ideas.

Clinton said...

Here are notes for the General Relativity class
(fall 2008)

Unknown said...

Dear Sir,
This is regarding the Special Relativity lecture 2 where you formulated the wave equation for the waves in an elastic string as a part of introducing the concept of a field theory. As I could understand you took the potential energy of each of the small springs (where each spring represented a model for infinitely small part of the string corresponding to a mass point) to be proportional to length^2. But I've learned that potential energy of a spring is proportional to extension^2.
It will be really great if you can explain this a bit.


johnl said...

Lecture Videos for Phy 27 General Relativity are now being posted on Itunes.
They are the 4th tab inside Modern Theoretical Physics.

PhysicsNovice said...

Dear Professor 15/01/09

Great idea to put your lectures on You-Tube,it provides a great opportunity for us lesser mortals!
My query regards Maxwells equations of electromagnetism.
Are there any lectures on Maxwells derivations of the above equations?

PhysicsNovice said...

Dear Prof,

You state that the Principle of Least Action is the bedrock of all fudamental physics theories, why is there a problem in the formulation of a sound quantum gravity theory.

Prashant Gokaraju said...

Is it possible to upload the general relativity videos any faster? There's only one video so far which was uploaded a week ago.

Rucko said...

prof Susskind, I have found an equation that describes gravity. as 2*c^3/hbar. This equation even takes care of the relative velocities, and mass increase. Please make an exception and examine my hypothesist. you only need to give one reason and I willbe satisfied. go to

Lenny said...

I would like to pose some questions concerning "The Black Hole War" - Susskind's last book. Assume Bob is somewhere well away from a very large black hole horizon and Alice is falling freely toward the horizon. Then in Bob's frame of reference Alice will encounter higher and higher temperatures as she approaches the horizon and will burn up. Of course in Alice's frame of reference, she feels nothing and notices nothing unusual. I have three questions about this situation, all strictly from Bob's point of view.
1) If Alice's spaceship has a digital readout on it that gives the temperature of the spaceship, will Bob see the readout go higher?
2) If Bob could observe Alice's face would he see beads of sweat forming and would he see her in obvious pain?
3) If ahead of time Bob had asked Alice to send him a message if whe were to start feeling warm, would Bob ever get such a message?
I don't think the answer to these questions is yes, but on the other hand don't they have to be in order for the world to be consistant in Bob's frame of referance?

Prashant Gokaraju said...

Add the videos faster!! 2 videos in 3 weeks!? COME ON!!

BLackIce said...

dear sir
i would like to thanks you for ur great work, i follow this blog from one of ur lect. in you tube
also i have a question
i read in wikipedia that enistine did not consider a twin paradox in his original paper in his original paper( i know the original version of the paradox) and he considered it as a consequence of the theory
is the paradox real or not and how mr Einestin consider it as a result of the theory
also i am interting in ur opinion in that
best wishes

Azaph said...

I'm Asaf from Israel. I watched some of your lectures about classical Mechanics on youtube, and I must say I very enjoyed them.

I also have a question.

It rose in my mind when you mentioned the hall effect.

Under the right conditions, applying an electrical field in a certain direction can cause a particle to move with a constant velocity in a perpendicular direction.
Does it mean that this particle gets all of its velocity at once as the electrical field is applied and it doesn't need any time or any space to accelerate? Sounds very odd, yet useful. Does it actually work this way?

Unknown said...

Respected Professor Susskind,
I have a question related to Lorenge Transformation and the invariant t2 - X2. Is there a geometric surface where this invariance holds? In other words is it possible to visualize the space-time?

wdogmx said...

Wonderful lectures - thank you! Here's a question: I have noticed, as I am sure many have, that several mathematical objects like the Kroneker delta, and the co- and contravariant matrices and tensors have appeared both in the lectures about Quantum Mechanics (and Field Theory) AND in your lectures on Relativity. On the quantum side there's everything from the simple two-slit matrix math to calculating particle motions in an electromagnetic field and things like spinors. On the relativity side, there the metric tensors. I have found myself wondering what if any connection there might be. Does the appearance of similar mathematics in the two subjects point to some deep connection?

robi said...

First of all i want to thank you professor Susskind for these lectures,they are really-really helpful, you are a great teacher.
My question would be:
When you construct the Action for a field in space-time. Why is the deviation from the real trajectory a one-variable fuction, namely f(xt) , and not f(x,t). Does is have any real significance?

Ethan said...

f(xt) is a shorthand for f(x,t). It does not mean multiple x*t. Also, x is usally a vector, such as a 3-dim spatial vector. So f(x,t) assigns each point in space-time either a number or vector value.

In later lessons you'll see a more compact notation.

Unknown said...

Dr. Susskind's lectures are great. Here are two comments of a general physics nature, and I am sure they fly in the face of accepted theory.

1. Things do not simply appear or disappear into thin air. (How can the universe suddenly appear out of nothing? Surely the universe must have already existed, but in some other plane or dimension, from which it emerged.)

2. Things do not suddenly happen for no reason. (Why does an electron suddenly decide to jump to a lower level, and release its photon? There must be something to trigger these events. Maybe the quantum foam.)

PS - My gmail account does not work, so I am using Bill's.

Robin Browne
Ottawa, Ontario

Unknown said...

One final comment, relating to number 1. above.

We can't just squish a star until it is the size of a pea (a black hole). Surely the matter is disappearing into space-time greatly warped and squished by gravity, but the matter is still there, and of a reasonable size, in some other dimension.

Robin Browne
Ottawa, Ontario

Unknown said...

In Einstien's General Relativity,"gravity" is described as the warping in spacetime fabric.The smooth spacetime fabric is assumed to be warped by "heavy" or "massive" objects which feel as gravity.This picture is completely different of Newtonian concept of Gravity.But Prof my question is that the concept of "heavy" or "light" objects come from the Newtonian gravity.How can it be used recursively to define "gravity" in General Relativity?If two objects one being very massive and the other being very less massive are placed in zero gravity they have actually no significance.It is the presence of gravity that makes massive objects "heavy" and less massive objects "light".
So during the explanation of origin of gravity in General Relativity, it is seen that the effect of gravity is already pre-existing.

Pollywog said...

I have a question that is about physics but not about the subjects covered in the videos. Why are the small objects floating in space in the same orbit as the International Space Station dangerous to the station if they are traveling in space at the same speed as the space station?

Perhaps this assumption that the objects are moving at the same speed as the station is incorrect.

Thanks for the blog and the videos.

wdogmx said...

Once again, thank you for these wonderful lectures. I have just finished watching #6 in the General Relativity series and have a question about parallel transport of vectors and curved space. Your explanation of angular deficit/excess as it relates to curvature is very clear, but I guess I am having a little trouble with the the idea of curved space. I understand that when one parallel transports a vector around a convex or concave 3D object such as a sphere, cone, torus or saddle, the vector does not return to itself and instead is “off’ by an angle theta related to the amount of curvature. But “holding the vector parallel to itself” is something that you do in the flat space in which the curved object (call it a subspace) is embedded. Vectors can only “wiggle” in the available dimensional directions and remain parallel in those same dimensions, no? A vector in the plane can only wiggle within the plane. There were a few questions about this from your students at the end of the lecture, but I didn’t find the answers that comprehensible. You mentioned that one doesn’t think about about keeping the vector “in” the curved space when doing this process. But then are you not parallel transporting the vectors in some idealized flat space “outside” the object (subspace) being measured for curvature? If so, how can one speak of a curved space? You are measuring curved subspaces in an idealized “exterior” flat space in which they are embedded, no? If space ITSELF is curved, then must not the vector must reside in that space and “curve around” with it? I am repeating myself, but it seems that if we are about to look at gravity as curved spacetime, according to this argument, we will have to think about some sort of idealized flat 4-space (Special Relativity) in which the curved 4-space of our universe is embedded. And I am pretty sure this is NOT what you mean.

Novice said...

Dear Sir,
I am going through your lectures on GTR. I am pondering on a the Riemannian Curvature tensor. You said that two of its indices defines a plane in which the vector gets rotated if it is parallely transported to itself and I think it is the tanjent plane. The other two indices defines the plane in which the vector is parallely transported back to itself. I am having difficulty in imagining this plane without embedding it in a higher dimension which I is not in the case of tanjent plane. Please help.

Kiran said...

Dear Leonardo

I was highly impressed by your video lectures on quantum mechanics and at a point,in following those lectures I got confused with the maths.So I decided to make a thorough study of LInear Algebra and Quantum Physics.
I went to the MIT-OCW site and downloaded all the recommended books and have been studying from them.R.Shankara, David Griffiths and Sakurai.

I am very serious in learning about the quantum world.Can you please guide me as to what to study and where?

By the by your lecture on the confusion beteen vectors and VECTORS got me writing this limerick

A vector,in general
is a nonarrow,though
born of an arrow,

In the abstract sense,
It is not narrow
It points the way,
to broader realms

if you read betwixt the lines
It is really,
a no-narrow

Kirankirti Chauhan

Novice said...

Dear Sir,

This is just a request. Could you please explain Singularity Theorems a little bit?

Thanks and Regards

Rucko said...

No answers or response from the Professor.

Mark Seery said...

A perspective of reducing the effects of relativity within an information universe:

Consider a train traveling between Nagoya and Kyoto at 300 km/h. Imagine there are two long-range radio transmitters, one at Kyoto and one at Nagoya. On the train is a handset capable of receiving two text messages at the same time when moving at 300 km/h. When the train is 3/4s of the way towards Kyoto, the transmitters at Nagoya and Kyoto transmit a text message at exactly the same time. The train being closer to Kyoto than Nagoya results in the handset receiving the text message from Kyoto before it receives the text message from Nagoya. The person who owns the handset looks at the handset and notices first the message that came from Kyoto, and then moments later the message that came from Nagoya. The handset owner might be tempted to say that the message from Kyoto was sent first, and an independent observer might be tempted to say that the owner's perception of which message was sent first is relative to the owner's frame of reference. Of course information network engineers and architects know that is not the end of the story when it comes to information networks.

In reality, the handset owner has suffered information loss. All the handset owner is really entitled to conclude is that one message was received before another, not that one was sent before another. Especially a handset owner that is aware that distance impacts the propagation of signals. However, that is still not the end of the story because in information networks we are able to create and recreate information. Each text message could have been sent with a timestamp created at exactly the same time by clocks that were precisely synchronized to each other, and to a clock on the train. Then the handset owner would be able to look at the source timestamps and know that both events occurred at exactly the same time, and in fact by observing when each message arrived know how far from each source the train was when the messages were received. Conversely, by keeping track of the time since the train left Nagoya (assuming uniform velocity and direction) the handset owner could have recreated the source timestamps of each message. Either by sending information with the text messages, or by recreating information after the text messages arrived, the handset owner would have sufficient information to correctly and completely process the question: were these two events simultaneous at their origins?

Let's take the case where the timestamp is appended at the source. Now imagine one of the text messages encounters noise along its way to the train and the timestamp becomes corrupted. Depending on how corrupted the timestamp is, a valid timestamp may still exist, but it may be inaccurate. As a result, the information is not accurate enough to correctly process the question: were these two events simultaneous at their origins?

Now suppose that the timestamp is sent in a separate message and it gets buffered along the way, and that the timestamp information is needed within a specific interval because the handset owner had a need to know whether the two events were simultaneous at their origins. Well depending on how long the timestamp was buffered or how long after the original message was received or how far from the train the source is, the timestamp may not be received within the interval required; in a timely way.

Information engineering and architecture can not change the fundamental laws of physics, but it can impact the relativity of information processing within different frames of reference by adding more information to a system. That information will be (in)accurate, (in)sufficient, and (un)timely. This reality means that information network engineers and architects have a significant role to play in the experience a network user has regardless of that user's frame of reference.

Unknown said...

Dear Leonard Susskind!
I am a physics enthusiast, and your lectures have brought me joy and fun.
Thank you very much for sharing this with us. I'm very impressed by your pedagogical skills.
Rolf Røsok, Norway.

arlesterc said...

I think I posted the following in the wrong area so I will post again here. Thanks in advance for any response.

Dear Professor Susskind,

First, I must thank you for your courses. I do a lot of physics 'reaading' on the Internet and your course is the best I've ever run into.

I have just been viewed the third session of the General Relativity course and have come to the section on tensors. It brought back to me a point another Internet physics lecturer had made when he was describing tensors. He said that a tensor was a machine that you put two vectors into out and got out a number. So for instance in the case of the metric tensor in flat space, if you put the same vector in twice, you get the length of the vector which is the same if you choose different coordinate systems. I have read and struggled with the concept of the stress-energy tensor and I would like to know what would be the vectors that would get fed into the stress-energy tensor and what would be the number that comes out? My guess is that the vectors that get put in are the four-vectors of an object moving in curved space-time and that if I put the same four-vector in e.g. feed the stress-energy tensor two identical four-vectors I will get a number which stays the same in different frames of reference. Is that correct? And if so what exactly does the resultant number that the stress-energy tensor spits out represent? Some type of invariant 'distance' in space time?

Thanks again for your courses.

SwaStassijns said...

Dear arlesterc,

you can look upon the stress-energy tensor as a machine with two slots. In those slots you put the four-velocity of an observer: if you put this in one of the slots you get the density of four-momentum as observed by this observer. If you feed the four-velocity in both slots the 'machine' grinds a bit and spills out the mass-energy density as measured in that observer's Lorenz-frame.

reza said...

thank you very much for your lectures on the web. I have a problem understanding your mathematical explanation of lorenz transform: here is my problem:

If in your explanation of Lorentz transformations we replace speed of light (C) with speed of sound (Vs) ( or any other speed) mainly ( Having Vs=1)
we can obtain Lorentz transformations where (C) will be replaced with (Vs).!!!!
so what didn't I understand
thanks again

johnl said...

9 video lectures have been posted for Phy 27 Eintein's Universe. But I understand there were 12 lectures. Does anyone know if the last three lectures will be posted to Itunes?

wdogmx said...

congratulation to the professor on his book win!

Prashant Gokaraju said...

Message for Clinton Lewis:

Are the Statistical Mechanics on your page ones written by Susskind? When will you be posting the rest of them?

Unknown said...

Dear Prof. Susskind:

Is there a method to compute invariants for a given linear or non-linear transformation ?

Thanks in advance

Vivek Dabholkar
Houston, Tx

denisv said...

If the Planck length is the smallest length we can meaningfully talk about, how come the spectrum of the position operator is continuous?

Setting Planck length = 1, I get the value 3.5 from a measurement, where is the particle? Is it at x=3? x=4? None? Both?

Unknown said...

Dear Leonard

I´ve been watching your lectures in Stanford available in Youtube about modern Physics. As always, Physics is many steps ahead from its everyday life consequences. In my humble opinion, the best ones are about enlarging the human perception and conception of reality. As an arquitect, I´m used to represent 3 or 4 dimensions in 2D projections of many kind. I would love to know how to represent stuff that happends in the current 11 dimensions predicted by string theory into 3d objects. It´s ok that, as you brillinatly said, we have to "rewire ourselves" to understand modern Physycs, and maybe arquitecture can help somehow, using matematics and computer graphics. Are you aware of anyone studying about it? I would be really glad if anyone can e-mail me:

Thank you and congratulations on your contributions to the humankind

Anonymous said...

Take two small masses, say 1 gram each, seperated by a very large distance. How large would the distance have to be for an accurate measurement of the force to violate the uncertainty principle? It seems to me that, for some large distance, the force would become so small that an accurate measurement of it would violate the uncertainty principle. Is this true?

johnl said...

The 8th lecture of Cosmos by Susskind has not been posed to iTunes. Does anyone know how to get it posted?


johnl said...

The 8th Lecture of Cosmology has been added to iTunes.

John said...

Are the Cosmology lectures online yet?? I can't find them. I guess they may only be on itunes now. Is that true??? Also I am trying to find an online type of masters degree in physics or applied physics or maybe a Phd. There are no schools around me that have master's or Phd's in physics. I currently have a 4 year degree in electronic engineering. I live in the southern part of the state of New Jersey. I have looked all over the internet and can find nothing. Would anyone know of a school that would have a masters or phd in physics that could be done in an online fashion. I don't want to have to travel anywhere.


John said...

I have been looking at a web site called millennium relativity. It is at, Is this good information or a waste of time. The papers seem pretty interesting. I haven't read all of them yet but I have just started to read the papers by the author Kristos Mavros.


John said...

Does anyone know anything about warp drive theory?? What happens if ship is standing still versus moving inside the warp bubble?? Is there an acceleration of some kind to a speed > c or is it an instant jump to a v > c. This may not be related to this forum but I think I heard that this forum was for general physics questions. If no one can answer my question I will post this to another physics forum. Does anyone know of some good ones????


Prashant Gokaraju said...


I dont think Susskind checks this page anymore.

Unknown said...

DR. Susskind,

let me ask a "stupid" question on classical field theroy. I can understand that electrodynamics can make good use of the filed thoery. But how to apply field theory in discribing a prtilce's moving? A partidle is a point but a filed is something spreading everywhere in spacetime.

Thank you!

Unknown said...

Questions from Cosmology lecture 4: 1. What fills the vacuum as the universe expands (and maintains constant energy density)? 2. Intuitively, it doesn't make sense that tension (negative pressure) would drive an accelerating expansion. How can this be?

My naive model (on a linear universe): consider a twisted string -- anchor one end, twist the other. Loops form, then loops on loops. Tension increases, like a coiled spring. Release the string, and it uncoils -- first the high energy, tighter coils, then the lower energy, looser coils. So -- the "universe" has tension (resists stretching when you try to pull the twisted string) but expands (maybe even exponentially?) as the tighter coils (short wavelength) then looser coils (longer wavelength) unwind.

Is there any possible physical basis to these imaginings? Underlying strings uncoiling previously hidden dimensions (and their wave packets) into the universe?


Unknown said...

I am ommair from pakistan. I want to ask about entropy, why it does not decrease. Can you send me link of introductory lectrues on special relativity.

cgryzgn said...

Im from turkey and Im watching your lessons for my modern physics class. Thank you, for telling relativity clearly.. Especially more than my teacher :)

Moses said...

Respected Dr. Susskind,

As being MS student, I am currently trying to review some very basic publications on Inflationary Cosmology. Before this, I have studied the courses of general relativity, cosmology and quamtum field theory. And your vedio lectures on general relativity helped me a lot for the better understanding of the subject.
I am emphesizing onto the WMAP aspects of of the Inflationary Scenario.
Can you please recommend me the related research oriented text books for this review work?

Moses said...

Respected Dr. Susskind,

As being MS student, I am currently trying to review some very basic publications on Inflationary Cosmology. Before this, I have studied the courses of general relativity, cosmology and quamtum field theory. And your vedio lectures on general relativity helped me a lot for the better understanding of the subject.
I am emphesizing onto the WMAP aspects of of the Inflationary Scenario.
Can you please recommend me the related research oriented text books for this review work?

johnl said...

Where can I find the lecture notes or Student's notes for Statistical Mechanics?

skyhqqk said...

How did Einstein calculate the
precession of the perihelion of Murcury and how do you do it?

Ansatz said...

I am currently a BSc(physics) student. Will you please tell me how would one explain the lack of locality in quantum mechanics.

Unknown said...
This comment has been removed by the author.
Unknown said...
This comment has been removed by the author.
Unknown said...

Dear sir, my question:
(ref: Relativity lecture) Does the minus sign for -dt^2 of the space-time come about from the fact that we measure time not from the perspective of light( i.e 0 proper time) but with respect to rest in our inertial frame? just like measuring length from the opposite end of the ruler backwards)? And I find it hard to treat space and time on a symmetric footing because our conscience always moves along the positive of time axis (never backwards). Also in four dimensional space-time x(t),y(t),z(t) is has meaning but for example x(z) or y(z) is abstract: just like a instantaneous cube has no meaning in reality (is just an abstract idea).
Kindly pardon if my doubt is silly. Thanks for your lectures

Love physics and love you

hans said...

In lecture 3 of cosmology you satd the age of the universe is 10 bn years
In lecture 2 you said it was about 13 bn years (usibg the H constant)
Could you please explain this difference


hans said...

why does e=mc2 does not apply to photons


Unknown said...

I'm a physicist from Colombia. I'm following your lectures at Standford in the youtube channel. You talk about notes on Quantum Mechanics. Please, could you give me the link to these notes ?


Callejero said...

I resently discover the Stanford couses on youtube and I am enourmously happy to be able to watch them. Thank you very much, Professor Susskind.
I am not a physicist and maybe my question is funny, but anyway I would like to ask it.

The universe is expanding and acceletaring, ok. The matter not bouded by gravity (galaxies clusters, that stuff) is getting far apart from each other becouse the very space between them is grown biger. Well, maybe it has not sense to say it, but... What's the diference if I said that all the matter in the universe is just Shrinking? That though came to me a few days ago and I'm having a little trouble to get it out of my head. Why should the matter be shrinking? I don't know. Why sould the space be getting bigger? The vacumm energy could not be a kind of force that causes the matter to shrink while not changing it's properties?
To me it sounds wrong what I'm saying, but I don't know how to reject the idea.

Thank you very mucho to read my question. Sorry for my english, I know it pretty bad. Grettings from Argentina to all.

Anonymous said...

Does anyone know if it's possible for there to be an uncertainty relation between General Relativity and Quantum Mechanics? In the sense that the effects of GR and QM can't both simultaneously be observed?

Thanks in Advance! (and I hope this question wasn't naive)

Abhishek said...

Hey guys...I was interested in knowing the wave nature of unpolarized light. Is the electric field vector in an unpolarized beam is in several directions at any given time? If so, then why dont opposite vectors cancel each other out? If not so, how is it that the electric field vector is cumulatively considered to be in all possible directions? Also, are the waves the vectors of which lie in different directions in phase with each other? I was hoping I could clarify some of those doubts here.

Unknown said...


I have a question regarding the magnetic field and forces on charges.If c harges are in rest, no magnetic field exist, but charges in motion i.e electric current will set up a magnetic field. If we have charges in rest we can imagine another reference system in motion relative to the first reference system. Here, a magnetic field will be detected and also a force F = qvXB because the charges have a velocity. There must be an error with the reasonning, because the forces cannot change with different reference systems.

Unknown said...


I have a question regarding the magnetic field and forces on charges.If c harges are in rest, no magnetic field exist, but charges in motion i.e electric current will set up a magnetic field. If we have charges in rest we can imagine another reference system in motion relative to the first reference system. Here, a magnetic field will be detected and also a force F = qvXB because the charges have a velocity. There must be an error with the reasonning, because the forces cannot change with different reference systems.

free electron said...

In the first lecture in the Special Relativity course, Susskind mentions that there was a previous course on special relativity/electromagnetism online somewhere. I can't find this course anywhere. Can someone help me out?

Thank you so much for these lectures, Dr. Susskind. I can't tell you how happy these make me.

cloya said...

Dr. Susskind,
I am taking modern physics (what they call here physics III). For reasons that are irrelevant to my question I am seeking additional resources for my education in addition to the class. I found your lectures on youtube about special relativity (which is what we are covering right now). However, they are the only ones I can find. On lecture #1 of those particular videos ( you make it a comment I would like to take advantage of. You say that you would go into teaching special relativity more in-depth if it were the first time you were teaching it to someone and that you have past lectures online that do this. From there you say that you will go over it more briefly and if anyone wants they can find your past lectures of "special relativity and electromagnetic theory" on the internet. Well I have turned the internet upside down looking for those videos that you referred to, so that I could get a proper intro to special relativity, and I have not been able to find them. Can you direct me to the specific site were I might be able to get a hold of these lectures?
Furthermore, the intro on this blog was very appealing as it referred to additional lectures which could be found at "://". I type this into the URL and get practically nothing! It takes me to the web site and it has no mention of you or your web lectures. All it has is a few books I could buy that are generally about physics. This is of no use to me. If you could please tell me where the promised lectures are it would be highly appreciated. Thank you for your time.

cloya said...

Dr. Susskind,
I am taking modern physics (what they call here physics III). For reasons that are irrelevant to my question I am seeking additional resources for my education in addition to the class. I found your lectures on youtube about special relativity (which is what we are covering right now). However, they are the only ones I can find. On lecture #1 of those particular videos ( you make it a comment I would like to take advantage of. You say that you would go into teaching special relativity more in-depth if it were the first time you were teaching it to someone and that you have past lectures online that do this. From there you say that you will go over it more briefly and if anyone wants they can find your past lectures (special relativity and electromagnetic theory) on the internet. Well I have turned the internet upside down looking for those videos that you referred to, so that I could get the full length introduction to special relativity, and I have not been able to find them. Can you direct me to the specific site were I might be able to get a hold of these lectures?
Furthermore, the intro on this blog was very appealing as it referred to additional lectures which could be found at "://". I type this into the URL and get practically nothing! It takes me to the web site and it has no mention of you or your web lectures. All it has is a few books I could buy that are generally about physics. This is of no use to me. If you could please tell me where the promised lectures are it would be highly appreciated. Thank you for your time.

Don Stevens said...

Have you seen the electron radius equation that links electron photon sphere radius to the radius (3/2)^1/2 times Planck length?
R2=electron Compton wavelength/4pi
R2=h/4pi mc
R4=electron photon sphere radius
R1=(h/4pi mc)^1/2(3Gm/c^2)^1/2
R1=(3hG/4pi c^3)^1/2
R1=(3/2)^1/2 (Planck length)
You have suggested the electron is not much larger or smaller than the Planck length. Gravitational blue-shift reduces Compton wavelength to 4pi(R1) size. When equal space contraction is included, Compton wavelength becomes 4pi(R4).
The electron, when gravitationally collapsed to its photon sphere size, has a radius value, slightly larger than the Planck length.

Don Stevens

Unknown said...

Dear Professor Susskind,
I am new to special relativity. So, this question must have been asked before. According to Lorentz transformation, if I travel to LA from NYC on a jet with a speed close to c, events I see happenging in LA are future events for my brother on the ground. So, I could tell my brother about HIS future event, say a lottery number. He can then make a decision based on the future, say buy a lotty ticket, and thus changing the course of future. But if he could change his future, what I saw on the plane can not be the future. Things do not seem to reconcile.

Anonymous said...

For cloya:

That link should open a list in your itunes with tabs at the bottom for lectures by Leonard Susskind on various things including special relativity. Hope this helps!

Nils/Hamburg said...

Dear Prof. Susskind,

congratulations and thank you for this great initiative. I watched your lectures with great interest and have the following questions on cosmology:

1. Could the apparent surplus of matter in the universe not come from an incomplete anhilation with anti matter, i.e there is still some anti matter out there, possibly beyond the horizon?

2. Is space itself a consequence of the inflation/expansion or ist the energy/matter expanding in an already existing space?

3. The term (adot/a)^2 represents kinetic energy, but (adot/a)^2 = constant means, adot is increasing, hence, shouldn´t the kinetic energy increase as well?

4. If the the universe is spatially flat and open (and since it began with a finite size during inflation) does this not imply that the universe is still finite with an edge, thus not homogeneous? And, if so, what is outside the edge? Empty space?



Aldo Mendes Martins said...

I'm very enthusiastic about the physics courses I have listened to so far. Dr. Susskind's empathy reaches to the level of poor enthusiasts like me who never had the opportunity to learn modern physics in university (as I am an engineer) and always had the curiosity to understand the world more deeply. Dispite of that, I have not been able to find the correct order in which I should do the courses. Can you post the order in which one should do all your online courses on modern physics? say, classical mechanics, quantum mechanics, etc. Thank you.

Pierre Asselin said...

A question about conservation. Einstein observed that it was impossible to distinguish acceleration from gravity. Am I right in understanding that this takes force out of the equation: F=MA and replaces it by gravity: G=MA. The mass of object 1 and the constant G is enough to obtain the acceleration of object 2.
So: what tickles me is this: An apple accelerates as it falls from a tree. "How" does it accelerate? what is the mechanical action that generates the acceleration?

Which brings my second question. If we were to say that in gravity, the apple's momentum changes without external force, and from there conlude that conservation of (linear) momentum is invalidated; Would that make things easier to understand, would it allow operations and hypothesis that are frobidden now because of conservation of momentum? or on the contrary would it make things so complicated that they would be unsolvable?
I am reading The black hole war, and enjoying every page of it. It is so simply explained and so clear. Of course nothing in your book suggested my questions.
Quebec City, Canada

jpc said...

I, admittedly, no nothing of the technical aspects of Physics. However, your book, The Cosmic Landscape, was as much philosophical as technical and I have the need to vent. So I do it here with no expectation of response.

“A scrambled hologram of an infinity of pocket universes” is how you describe where the current state of String Theory is leading us. Yet at the end of The Cosmic Landscape, you repeat your agreement with Laplace that, when it comes to god, “I have no need of this hypothesis.”

I don’t know, anything that appears in “infinite” numbers, especially universes, sounds an awful lot like a definition of god to me. It’s astonishing that in the 21st century brilliant scientific minds like your own continue to inveigh against a concept—god—which you then fail to define! No, Mr. Susskind, no one with an active brain continues to believe in the personal god of my—and probably, your—youth. But, with your thralldom to 21st century slide rules, you strip awe, wonder and mystery away from who we are and where we come from. If there’s a mathematical formula which can describe “reality” (even one with infinite solutions describing infinite “realities”!) that’s all the evidence you need to instruct the rest of us that we are random, purposeless mutants per Darwin—or at least per those acolytes of Darwin who seem to have an almost fetishistic need, eerily similar to their fundamentalist religious counterparts, to prove they are right that god is dead.

The Bible/Torah/Koran all use the word “ineffable” when describing god. The 3 or 4 pages you devote at the end of your 380 page book to what you call the “measure problem” (i.e., the “infinite” problem in your “infinity of pocket universes” solution) might have been better spent meditating on the meaning of the word “ineffable.” However much you dissed Paul Davies and his “religious” slant in the books he’s written, Mr. Davies, at least, has begun that effort.

I rant because, to me, when our intellectual leaders propagate a paradigm which insists on purposelessness within creation, we have nowhere to turn other than to sating to excess the bag of bones we inhabit. Though you may agree with the Christopher Hitchins and Richard Dawkins of this world that their “ethics” are as profoundly felt as those of any non-materialist, their position just doesn’t make any sense to me. And I tell you that from profoundly felt personal experience. Perhaps the world to you looks peachy-keen, but I would suggest that we are in profound need of a new paradigm.

Unknown said...

there's something i can't understand about uncertainty principle: momentum and position aren't really determinated with infinite precision, or it's only men that can't measure them at the same time with infinite precision?

vk said...

I love your lectures Leonard, thank you so much for providing such an amazing resource!

Philip said...

Um, I'm not sure if my questions are sensible, but I'll ask them anyway.

I was watching the course on Special Relativity and I got all the math. The hyperbolic functions certainly made sense, but this is the first time I'm encountering them. My question is where did they come from? It looks as though we just *defined* a new set of functions that suited us. Is that 'allowed'? I mean, couldn't you have also defined another (different) function that also gave the same results?

Why don't the usual trigonometric functions work? And how did the idea to work with hyperbolic functions come about?

michele said...

Dear All

I've read the wonderful Susskind's book "The Black Hole War" and the fundamental Witten's paper "ANTI DE SITTER SPACE AND HOLOGRAPHY". I know that the holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation by Leonard Susskind.

In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon, such that the three dimensions we observe are only an effective description at macroscopic scales and at low energies. Cosmological holography has not been made mathematically precise, partly because the cosmological horizon has a finite area and grows with time.

The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insight was that the description of all the objects which have fallen in can be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory. I'm a studious of mathematics and theoretical physics, principally string theory and Number Theory. My researches involves the possible mathematical connections between some sectors of string theory and various sectors of number theory. I think that various equations concerning the string-theory interpretation of the olographic principle can be connected with some sectors of number theory (Ramanujan's modular equations, Fibonacci's number and Aurea ratio). Thanks,
Michele Nardelli

Seer Magnus Prime said...

You must consider the concept of The Calabi-Yau Manifolds in your estimation of the Fabric of the Universe! There is not merely one, but, numerous representations of them! The spaces between the interlacing of the manifolds, allows the passage of fluxes that power the field! Doc

Unknown said...

Physics has some explaining to do. For instance: Since there was a Big Bang, then where did the Big Bang come from. If space itself is being spontaneously created by Inflation or Expansion, then where is this space coming from.

Not being a religious person, it is obvious there is some kind of parallel here: If God created the universe, then where did God come from.


Seth Olsen said...

Dear Prof. Susskind,

Many thanks. I enjoy your video lectures.

I just watched the one on no-cloning. I have seen a few expositions of the no-cloning theorem, including yours. During a seminar the other day, I think I obtained some new insight into it and wanted to ask you:

Is the real reason behind no-cloning that the entropy is extensive?


Seth Olsen
U. Queensland

Unknown said...

Thank you for the opportunity Mr Susskind. There are a lot of references to relativity theories and quantum mechanics in this blog. By a new paradigm understanding quantum mechanics is relegated to a secondary theory, with no relevance to physical reality itself. The theory claiming this is Empirical Relativity Theory. It also gives reason to, yet transcends, special and general relativity theories. As this is a blog I won't go into detail, but google it. It is in keeping with the latest dark matter theories questioning the relevance of General relativity being spruked in the latest New Scientist. Check it.

Jardbird said...

Dear Prof. Susskind,

I'm from Germany and enyoed your Lectures about Physics very much (so please excuse my bad English)

Maybe this is the right Place du ask two Questions, which bothered me now quiet a Time:

1)Is it right that due to Inflationtheory each Point in Space inside our Horrizon has an equal Horizon arround itsself? And if we suggest that the Universe is isotropic each point has an equal amount of Mass arround it and due to that no gravitational Pull in any Direction (in great scale, not local) If there were no Vacuumenergy (Darkenergy) how could there be a universal Contraction through Gravitational effects then?

2)If seen a Simmulation of a fast spinning massive Object and its gravitational Waves, they first stayed in nearly symetric tught Spirals. But after a while they evolved in a Barform with two spiral Arms at its Tips, slowly rotating around the Center. This reminded me instantly to the Shape of a Bargalaxy. Is there any Relation between the Rotation and Masses of Blackholes in the Center of a Galaxy (and so its emited gravitational Waves) and the Shape of the Galaxy itself?

Thank you very much for your very interesting Lectures, Iwould love to see mor of it.

siddharth said...

its a great opportunity and an honour to learn from you professor sir.
I am a student of computer science so no physics courses in college.
Thank you for these lectures.
I wanted to ask would you please do QFT in near future ?

Brandon A. Figueroa said...

Hi Professor,

I am currently taking a Modern Physics course at UC Santa Cruz. I've just started watching your lectures and enjoying as well as learning a lot from them.

I have one specific question about special relativity. This was one of my homework problems and I had the hardest time (go figure) trying to figure it out.

Here it is: In the laboratory frame, a particle with mass m is moving with constant velocity v=(square rt 3)/2 toward a similar particle with the same mass but its at rest. What is the speed of the inertial frame (relative to the lab frame) in which the total momentum of the system is zero?

If you can help that'd be great.

Unknown said...

First, I must add my thanks to Prof. Sussking for the wonderful lectures he has posted.Of the dozen of books on gen relativity I have seen, he was the first to derive the Riemann Curvature in a convincing manner.
I think I speak for many in wishing for a course in String Theory. If anyone can explain this convoluted topic, Mr. Sussking can.
Finally, several authors have stated that the best way to think of an electron spin is to imagine a particle that takes two rotations to return to the original position. I think I went crazy trying to visualize such an object until I learned from the QM course that this is abstract vector space which has nothing or very little to do with pointer space.
Finally, how about a course in Quantum Field Theory?

Unknown said...

In response Brandon:

I think it is necessary to split up the original momentum into two equal portions which will cancel out. If you give the inertial frame a velocity 1/2 of the sqrt3/2, then in the inertial frame the two particles will have the same speed but moving in the opposite directions, thus cancelling out the momenta. Maybe?

Brandon A. Figueroa said...

In response Henry,

Thanks Henry. That makes sense. I was confused because since there were no external forces and the two particles are only interacting with each other. So I guess I was naive enough to think that the momentum was zero in both frames and that the velocity of both frames should be v=(sqr rt. 3)/2 relative to each other.

But thanks a lot Henry. It helped.

Piotr said...

Dear Professor Susskind,

I am writing to you from Poland. I have just been listening to you lectures on Special Relativity/Classical Field Theory. I like very much the mechanical example you use to explain the sense of a field theory - namely that of a string. I have three questions in this connection:

1. Can this analogy be extended on fermionic fields? I personally think it is not possible, but could you comment on that?

2. You explained that the conserved quantity (i.e. the charge) resulting from the invariance of the Lagrangian with respect to global gauge transformations corresponds to the angular momentum of the string. Is it possible to extend this analogy on the case of LOCAL gauge transformations? What would be then the mechanical counterpart of the gauge field A_{\mu}?

2. Would the quantum field theory be connected with quantum-mechanical string in a similar way as classical field theory does with classical mechanics of the string?

Looking forward to your answer


Javier said...

Hallo all the people in the blog. Only want salute all. I like very much the Susskind's Lectures and the blog.

Javier said...

I'd like the opinion of Mr. Suskind and other persons within membership to blog about the following question:

We know that an accelerated charge on a frame of reference results in an electromagnetic field in that frame of reference we call acceleration field or radiation field.

For the equivalence principle a state of acceleration is quite equivalent to a gravitational field. So if we place a charge at rest relative to a frame of reference where there is a gravitational field at a point away from the charge (where the field almost non-existent) we can measure electromagnetic field radiation is given by the familiar equation for the radiation field. Specifically, in the case of the electric field, using Gaussian coordinates we have:

E = (e / c ^ 2 * | R | ^ 3) * R x (R x g) --- R, g vectors

R: distance betwen the charge (source point) and the field point.
g: gravitational aceleration (gravitational intensity).

Where we have assumed, of course there is no relativistic speed between load, located at the source point and field point. It is as if the movement was to begin at any time with an acceleration g. We have also assumed that the gravitational field at the field point is zero, which is not necessary, it is possible that the field exists at that point but has a different value. In this case we would have:

E = (e / c ^ 2 * | R | ^ 3) * R x (R x (g1 - g2))

Note that we can write this as follow:

E = (1/c^2)*Eo x (R x (g1 - g2))

Where Eo is the electrostatic electrical field due to the charge e.

g1 Where would the gravitational field at the source point and g2 be measured in the field point.

This equivalence: a----> g must be correct if the charge is not "falling" in the gravitational field, ie must belong to the reference from which the gravitational field exists.

I await your opinions.

PD. Sorry for my bad english

AlexanderM said...

Whether probably to consider light
(photon as particle or as wave )
as 2D object in 3D space?
how You think? I guess, that light it is geometric representation of material 2D objects in 3D space (4D space-time) and it is a fundamental property of light.

John said...

I have a naive question, related to cosmology. I don't know the math but my understanding is you can take pretty much any object and compress its volume to an extent that it becomes a black hole. Assuming that's accurate, when I hear about looking back in time and the universe is seen to get smaller in volume, its seems like at some point the threshold would be reached where it would become a black hole. I never hear about this so where am I going wrong ?

P. Gokaraju said...


Unknown said...

Prof Susskind
I very much enjoyed your lectures on QM. In your last lecture on QM you never got to angular momentum.
Did you ever get to it in General Relativity? if so, in which particular lecture?
Sincere thanks

MG said...

In lect.10 of the QM course, Dr. S. mentions a course on QFT.
Does that exist?

Unknown said...

Thank you Dr.Susskind for allowing everyone to take part in the world of physics, and not confining it to ivory-tower institutions. It is truly special to be able to learn from a scientist of your caliber. As a university student in Canada, I have paid as much as 700 dollars per term to take part in "distance education" courses which offer lectures, and professor access in much the same manner as you do here, for free.

dantes1 said...

I've been curious lately if anyone has done complex functions on cartesian coordinates?
Wouldn't dy/di e^i=e^i at b=1? The property that ln i twice differentiated is real might then be useful.


gab said...

Loved the book! Have a question about the Planck area and vol.

I calc'd the area of a black hole of M=Planck mass to see how many Planck areas (i.e., bits of entropy) would fit. It came out to approx 27-28.

Reasoning that (1) it should be a whole number, and (2) should be in the form of a regular polyhedron (1st approximation = 20, an icosahedron); concluded that:

(1) Diameter of that black hole should be the square root of 20/pi...or about 2.5 Planck lengths;
(2) A Planck Area should be an equilateral triangle vs square (and thus have a value approx 1/2 the established value); and
(3) A Planck Vol should be a tetrahedron vs cube(and thus have a value approx 1/8 of the estab value).

gab said...

Loved the book!
Have a question about the Planck area and vol.

I calc'd the area of a black hole of M=Planck mass to see how many
Planck areas (i.e., bits of entropy) would fit. It came out to approx

Reasoning that
(1) it should be a whole number, and
(2) should be in the form of a regular polyhedron
(1st approximation = 20, an icosahedron);
concluded that:
(1) Diameter of that black hole should be the square root of 20/pi...or
about 2.5 Planck lengths;
(2) A Planck Area should be an equilateral triangle vs square (and thus
have a value approx 1/2 the established value); and
(3) A Planck Vol should be a tetrahedron (and thus have a value approx
1/8 of the estab value).

princessleila said...

Hi Mr. Susskind !

I watched your (great) course, and felt like asking:

Would'nt it be simpler if one considere time as imaginary, to automatically have dx-super-mu times dx-sub-mu equal to -dt²,dx²dy²,dz² ?

We could even write dX² = -dt²,dx²dy²,dz²

is it a bad idee?

And also, i would like to know what do you, and other physicists, think of Jhon Cramer's Transactional interpretation of quantum mecanics?

It looks like an interesting aproach, but i wonder if physicists think it IS an interesting aproach. Does it solve "wave function collapse" difficulties and paradoxes as he claims?

Thank you a lot for your course, and for your (proper) time ;)

michele said...

Dear Prof. Leonard Susskind,

I'm a studious of mathematics and theoretical physics. I'm now writing because I'ld like to know from You the most important papers related to the fundamental discovery that there would be no "information loss" in black hole quantum evolution.

Thank You for your availability

Michele Prof. Nardelli

dantes1 said...

Has anyone ever set hv=mc^2 to solve for the mass of a photon? The number seems reasonable and its such a simple thing to do I've been surprised that I havn't seen it.

dantes1 said...

I think I found a way around the Heisenberg uncertainty principle. Assuming that a solid was isolated from its environment in a vacuum, and a matter wave of say neutrons were shot into the corner of the sample, preferably the corner of the corner atom, I believe it would be possible to reorient the electrons of the remaining atoms of the sample by a process akin to van der waals attraction. A neutron beam with a very short wavelength would ensure that the electron would be forced to be attracted to the largest extent. There's the possibility of domains being established within the sample that would prevent the ideal alignment I'd like throughout, but that's a problem for someone else.


Rich R said...


If the age of the universe is approx 14 Billion years, how would the age estimation differ at different locations (space-time coordinates)? Is the age of 14 Billion years universe-wide, or could it vary significantly as function of location?
Thank You

Rich R said...
This comment has been removed by the author.
Unknown said...

In lecture 4 of special relativity, the relation between symmetries and conservation laws are reviewed. Interestingly, a symmetry is defined there as a transformation of coordinates $q_i\to q_i+\eps f_i(q)$
that does not change the Lagrangian. Later, the derivation relies on the fact that
$\sum_i (\partial L/\partial q_i f_i(q))=0$.

But when one says that the Lagrangian is invariant to a shift in coordinates, doesn't it mean that the \dot q_i must also be replaced by $\dot q_i+\eps \dot f_i(q)$?

So is a symmetry defined by the fact that the Lagrangian doesn't change when the coordinates are shifted, but the velocities are left intact?

Joe said...

Hi Prof. Susskind,

I have a question regarding GR. You use Hyperbolas as constant acceleration frames but I believe you did not show this in the course. Could you direct me to a Lecture or reference that proves this?


Joe said...

Hi Prof. Susskind,

I have a question regarding GR. You use Hyperbolas as constant acceleration frames but I believe you did not show this in the course. Could you direct me to a Lecture or reference that proves this?


Unknown said...

Dear Leonard Suskind,
I'm a beginner in physics but I'm trying to learn more as I have time for it.I have a question,I think it's simple but it's very important as you told in your lessons;it's about lagrangian and least action principle.As I understood,we minimize (F-ma) over space-x,and also we can minimize (T-U) "lagrangian" over time-t,then we can find the trajectory of motion.Can you give me suggestions for understanding this principle more deeply?From all possible perspectives?
is it a kind of symmetry principle?
Thanks for everything,thank you:)

Unknown said...

And why nature minimizes these actions rather than others?Why nature is so mean?Is all possible actions for any system can be minimized?and found the trajectory of this system?Do you have any book recommendations for this?I want to understand this deeply really because it seems to me wonderful...

Sage said...

I was looking at your lecture #1 on special relativity to learn more about physics to
help build a matrix and noticed the drawing of time is different than the way I have used it.
Could you provide a piratical equation with the answer, that show two different times (t and t prime) or a link to a lecture that shows it being used to solve a real world problem

dk081867 said...

have you ever considered that the universe is nothing more then an infinite atom and contained within the quantum scale it is nothing more then a replica of the body of the universe.the only difference being between a particle,photon,electron etc would be the same difference there is between a hand,eye,hair or a foot but when looked at as a whole an entire body appears.

Unknown said...

Dr. Susskind,

I had to laugh during one of your lectures, when you said that you "have to be in two places at the same time".


Thanks for your wonderful lectures.

Ron Brown
Ottawa, Ontario

Unknown said...

I have a question about Special Relativity that has bothered me for some time, concerning "time dilation" ....

It seems to me that the only reason for the space-time diagram in STR is to DEFINE velocity - with the definition being translationally invariant in both dimensions once a grid is defined. Velocity requires two parameters for definition; (x,t) or (x, theta) etc.

However the space-like axis seems to me an existence statement - to that extent, it is unique (moves vertically up the time axis, with coordinate points moving along it according to their world lines.

(STR seems to avoid this definition of velocity; rather considering it to be a single point in a Riemannian manifold).

To this extent, in a space-time diagram, c is just another velocity (with the ST diagram divided into time-like and space-like regions) - and all velocities are independent....

so far so good: x = vt = (x/t)*x
given a particular mesh...

In particular, if c is linearly independent of v, one can create another diagram with c vertical and v horizontal. One can then postulate an invariant "t" so that an invariant "length" is defined by c*Tau. Then one can postulate a "t'" which varies the "length" along the velocity axis linearly so that this length is v*t'.

Thus - the Time Dilation equation.

Now comes the rub - I'm thinking that the second diagram is actually an E-K diagram, with all parameters replaced by energy-momentum equivalents. So c*t in the second diagram represents an invariant "light stick"... multiplied by a "density p=1", it gives m0= p*c*t, the rest mass of a particle, with different particles defined by different values of t.

This is then turned into a different definition of velocity defining inertia -

v = c*sqr(1 - T'^2/T^2) (p cancels out) Note that although the value of v changes according according to T (which I have capitalized to emphasize that this is not to be identified with "time")

This is consistent with the relativistic definitions of momentum/energy, but NOT of space-time (which remains cassical)

Since interactions are instantaneus along the space (x) axis, this means that QM interpretations of time (at its most fundamental level) is that of a "dummy" variable, showing the progress of an interaction (basic barrier problem), - like a steady-state convolution integral - rather than "time" in a space-time diagram (I'm also thinking Klein-Gordon equation) - the context of the discussion here is lineland...

This implies a particle definition of the Universe - no fields, so there must be a conceptual model with all velocities = 0, all photobs absorbed (if the universe is infinite, there is no center of mass = "observer") - and all particles (including photons) are modeled as "light sticks", with approriate relations to the de Broglie equations.

Anyway, I'm sure you can see where I'm going with all of this - I have worked out analyses for myself from Bergmann (especially the Lorentz transforms)... but are impossible to write in text....:-)

(And, of course, it depends on whether one postulates c as "special" a priori, or assumes Maxwell's equations, subsuming mass/energy in the permittivity/permeability constants)

There is something I'm missing here, but I'm thinking that if the universe it to be described in particles, then gravity has to be described in terms of statistical mechanics (a particularized version of Dirac's "sea")

If anyone is interested in this issue, I would like to discuss the issues, but can't here - email me at BuleriaChk "at" aol "dot" com

Unknown said...

I wrote a comment about "Time Dilation" in which I opined that the "T" in the equation:

v = c*sqrt[1 - (T'^2/T^2)} could not represent time, since it actually was the definition of rest mass for a specific particle (e.g., m0 = pcT for p=1, m'=pcT')

It seems to me that specification of the value of c sets the mesh of ALL moving frames, in the sense that x(c) = c*t(c), here x and c refer to orthogonal space-time. Once c and either x or t are specified, the mesh applies to all observers' frames, contradictiing the idea that "time slows down, space contracts", etc.

For me, it meanst that the "time dilation" equation is simply wrong; rather, the equation above establishes v as a function of c and the relationship of moving and rest mass.....

If this is true, and relativity expresses all particles in terms of "light sticks" (including photons), consistent with QM via the de Broglie relations, then I think it calls the whole underpinning of GTER into question, since acceleration can be considered a world line curvature described by piece-wise continuous Feynman diagragms..

I don't know if any of this makes sense to anyone, but if anyone wants to discuss this, drop me a note at "BuleriaChk "at" aol "dot" com, and I'll discuss it further here on this forum.. there are a LOT of moving parts.

(Fun exercise - take the Lorentz transforms - from Bergmann's analysis, and set x = ct; I find the fact that this gives the "time dilation equation (one equation, not two) significant and interesting in the above context)

On the other hand, I may "not even be wrong", as per Pauli...:-)

Unknown said...


Never mind, by trying to write this stuff down, I realized that for me my current issue is whether the fine structure constant (in the context of gravity) can be a function of position or not....


v = c*sqrt[1 - (T^2/T'^2)], T'>=T

(I ALWAYS write that wrong, dammit)

Unknown said...

Thanks a lot sir for your lectures.
I am pursuing B.Sc. physics honors. I do not have any teacher. But now, just because of your precious lectures, i am able to think logically as well as i got 96 percent marks in my exam. Even my performance in my practical exams was excellent. I topped in exams.
The credit of my success only goes to you. Thanks for every guidance you gave me via your lectures. Thanks for giving us a platform to contact you.

you are the best teacher for me.

Unknown said...

Absolutely. Even people who are not able to express themselves in the English language, as you are.

You have proven that you deserve our respect.

I always listen to your lectures.

Well, one day, we should prove the quantum reality of "cold fusion", or in fact a whole another "chemistry of the nucleus". But this is an extremely difficult problem.

I believe that it has to do with "quarks" or "strings" and Planks constant.. but I will never give up :-)

Sincerely, Rob.

LSB said...

Dear Leonard,
I discovered these lectures by pure chance on you tube. I have many questions but for the moment I would just like to say that you are truly an amazing teacher. I have followed your lectures in General Relativity, special Relativity and Quantum mechanics and have understood everything without having to make any notes. This is not because the subject matter is easy but more to do with your ability to explain the concepts. I welcome the many jokes that come in now and again. I studied at Imperial college London many years ago and feel privileged to have been in Seminars where the late Professor Abdus Salam used to be present on occasions. Is there any way that your lecture notes could be accessed/ published or is there a possibility that you may write a book based on your notes? I would mark Christmas 2009 as the year where I truly understood General Relativity due to your lectures . I am hoping the next few years will prove fruitful.I only wish I could be physically there.
Kind Regards

sdcojai said...

Ditto to LSB's comment on January 22. THANK YOU Professor Susskind for these invaluable lectures!

Unknown said...

Hi All,

I am currently auditing Professor Susskind's lectures on Quantum Mechanics and I am thoroughly enjoying them.

I have had some exposure to the Schrodinger formulation of Quantum Mechanics; however, Professor Susskind is presenting the Dirac formulation.

How are these two formulations related? Will I find that they both merge or that perhaps they are connected at the hip?

Thank you,

Unknown said...

Dear Professor Susskind,

I too would like a copy of the notes accompanying your lectures.

And very sincerely, thank you for enabling me to finally, after many, many years, understand the basics of Physics.

I wish that I could continue for another 64 years to get beyond the basics!

There is beauty in Physics.


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