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
http://www.learnoutloud.com/Catalog/Science/Physics/Modern-Theoretical-Physics/23022
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
Blog Archive
124 comments:
I think its great that you've started this blog. Looking forward to seeing the Q&A.
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?
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
Here's the location of scans of professor Susskind's lecture notes for the current Special Relativity and Field Theory classes:
http://web.mac.com/clinton_lewis/Special_Relativity/Lecture_1.html
Heres' a dramatic representation of a vector field.
http://sfports.wr.usgs.gov/wind/streaklines.shtml
This display shows the SF Bay wind pattern as "streaklines".
Dr. Susskind,
Can you recommend a reference(s) book.
Thanks,
JH
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
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
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
Mike
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.
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.
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
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.
Sincerely,
Chas Egan,
Australian National University
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
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.
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
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
Thus
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
Also
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.
Mike
Oh, correction. Actually, v=qdot, so
partial_q H = -partial_q L
for all q. Sorry about that. -Mike
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}
Mike
...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.
Mike
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:
http://dsp.ucsd.edu/~kreutz/PEI-05%20Support%20Files/complex_derivatives.pdf
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])
Ethan
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?
May 12 lecture notes are up!
http://web.mac.com/clinton_lewis/Special_Relativity/Lecture_5.html
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 YahooGroups.com 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
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
The February 18 QM note page now has a pointer to additional student notes:
http://web.mac.com/clinton_lewis/QM_Spring_2008/Lecture_6.html
Notes are also published for the May 19 class.
http://web.mac.com/clinton_lewis/Special_Relativity/Lecture_6.html
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.
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?
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?
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
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?
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.
-Mike
Hello,
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
Regards,
Cormac O'Raifeartaigh
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.
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?
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.
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.
Thomas,
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.
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?
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.
Thomas,
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...
Thomas,
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!
f=m*c^2/h
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.
Dr. Susskind.
Duh, posted this reposnse in the wrong post originally. What can I say..
Anyway,
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.
Regards,
Colin Bembridge N.F.D.A.A.
Toronto, Canada.
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.
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.
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?
Thanks
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!!
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?
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,
Jim.
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?
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?
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:
http://web.me.com/clinton_lewis/Special_Relativity/Lecture_1.html
and for the Fall 2007 Classical Mechanics lectures:
http://web.mac.com/clinton_lewis/Classical_Mechanics/All_Lectures.html
and for the January 2008 Quantum Mechanics class:
http://web.mac.com/clinton_lewis/QM_Spring_2008/Lecture_1.html
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?
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
fisicadivertida.ning.com
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?
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?
Those who attend Professor Susskind's GR class can find his notes at:
http://web.me.com/clinton_lewis/General_Relativity/Lecture_2.html
can someone point me to the videos of the GR class?
I download and view the lectures from Itunes. Go to Itunes U then to Stanford.
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?
In the past they have posted the lectures in a group sometime after the last lecture takes place.
Here's the location of scans of professor Susskind's lecture notes and student notes for the current General Relativity classes:
http://www.welkinsky.com/General_Relativity/Lecture_4.html
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.
http://www.nationalpost.com/news/story.html?id=859565
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).
Please find General Relativity student class notes and scans of Prof. Susskind's lecture notes on
http://www.welkinsky.com/General_Relativity/
Clinton Lewis
hi
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.
Hi,
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...
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?
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.
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
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
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.
When are the general relativity lectures going to be uploaded to itunes!!?? I've been checking it constantly for weeks!
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?
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?
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?
Thanks.
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.
Here are notes for the General Relativity class
(fall 2008)
http://www.welkinsky.com/General_Relativity/Lecture_2.html
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.
Thanks.
Lecture Videos for Phy 27 General Relativity are now being posted on Itunes.
They are the 4th tab inside Modern Theoretical Physics.
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?
regards
PhysicsNovice.
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.
Is it possible to upload the general relativity videos any faster? There's only one video so far which was uploaded a week ago.
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 rucko.com
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?
Add the videos faster!! 2 videos in 3 weeks!? COME ON!!
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
so
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
Hi,
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?
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?
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?
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?
Robi,
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.
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.
Sincerely,
Robin Browne
Ottawa, Ontario
613-276-2139
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.
Sincerely,
Robin Browne
Ottawa, Ontario
613-276-2139
Dip
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.
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.
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.
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.
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
kirankirti@gmail.com
Dear Sir,
This is just a request. Could you please explain Singularity Theorems a little bit?
Thanks and Regards
No answers or response from the Professor.
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.
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.
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.
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.
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
Reza
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?
Thanks
congratulation to the professor on his book win!
Message for Clinton Lewis:
Are the Statistical Mechanics on your page ones written by Susskind? When will you be posting the rest of them?
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
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?
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: woelzm@gmail.com
Thank you and congratulations on your contributions to the humankind
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?
The 8th lecture of Cosmos by Susskind has not been posed to iTunes. Does anyone know how to get it posted?
Thanks
The 8th Lecture of Cosmology has been added to iTunes.
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
I have been looking at a web site called millennium relativity. It is at http://www.mrelativity.net/Default.htm, 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
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????
John
Try physicsforums.com
I dont think Susskind checks this page anymore.
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!
ZW
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?
Thanks.
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.
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