James R has gotten to the heart of the matter here. The reason why thermal energy contributes is because there is no frame where all the particles in the brick are at rest. This is an essential feature of the fact that there are multiple particles in the brick, each with a different rest frame. Because of this ambiguity, the best "rest" frame you can find is the frame where the whole brick has zero total momentum. However, many particles are still moving in this frame and while their momenta cancel, their average energy (temperature) does not. As I mentioned earlier, on a technical level this can be traced to difficulties with defining a global proper time for multiple particles.
That is a completely sad argument - I am very dissapointed in you. Let's take another example. Let's imagine 100 particles in a volume of space zooming around with great kinetic energy. These particles as a bunch make no greater gravitational effect than if they were all at rest when we look at each particle individually. Now wrap a box around these particles. What is the mass of the box (including everything contained within) assuming the box is essentiall massless when compared to the mass of the particles (assume very heavy particles!). This is the same as your thermal energy BS in which you claim just because no single frame can be used to analyse the situation, then the mass must increase! No... if kinetic energy doesn't add mass to any frame, then it does not matter that you cannot switch to any one rest frame for these atoms with kinetic energy that we just relabel as thermal energy. These are claims without rigorous mathematics. I do not accept these claims, I've seen them elsewhere, I mentioned them above - I already know and you are not going to convince me of anything until you show exactly how kinetic energy in one case cannot contribute to gravity but it can in another case. That is BS and you should know it, but you never think for yourself...
Aer, I would ask you to consult the book Gravitation by Misner, Wheeler, and Thorne pp. 139-140. In this book, as in most any other book on GR, they derive the stress energy tensor for a perfect fluid (ideal gas). The stress energy tensor is as follows: T = p g + (p + rho) u * u * denotes tensor product g is the metric tensor u is the fluid's four velocity p is the pressure in the fluid's rest frame (rest frame defined as frame in which fluid has no total 3-momentum) rho is the density of mass-energy in the fluid's rest frame The authors of this book, each of whom is a famous researcher in GR, have this to say at the bottom of p. 140, "However, for a general perfect fluid, density rho of mass-energy as measured in the fluid's rest frame includes not only rest mass plus kinetic energy of particles, but also energy of compression, energy of nuclear binding, and all other sources of mass-energy." I would reproduce the derivation here, but I don't want to try and type math formulas into this interface. I'll try to locate an online source so you can immediately see for yourself.
In Sean Carroll's lecture notes that I posted earlier, the stress-energy tensor is described on pp. 28-29.
Also, the book by B. Schutz, "A First Course in General Relativity," has an excellent discussion of the perfect fluid.
I will consult as you request, but first let me comment on what we have here. First of all, the stress-energy tensor contains non-gravitational energies, does it state which energies these are? Second, an ideal gas does not exist, it is only an approximation. I had to point that out to one Quarkhead on a completely different matter in which he was using an ideal gas to prove something. Let me just clarify for everyone what an "ideal gas" means. It means that the internal forces are negligible and in fact are not taken into account. So when claiming something is an ideal gas, you are only working with approximations.
Aer, you will find that defining the gravitational energy of a gravitational field in GR is difficult and cannot be done in general. What we can say is that the gravitational field can affect (via the presence of g in the stress energy tensor) the way matter contributes to the field. However, the fact that gravity contributes to itself (via gravitational energy if you insist) is contained in the nonlinear left hand side of Einstein's equations (the Einstein tensor) as much as in the appearance of the metric in the stress energy tensor. To clarify, the non-gravitational energies are described by rho the mass-energy density. This is the energy the stress-energy contributes in the abscence of gravity (g is special relativity metric) and thus must be by definition the non-gravitational contribution. Also, I am perfectly aware of what an ideal gas is, nor do I claim that anything in the world is an ideal gas. That being said, the approximation of an ideal gas is often very reasonable in applications of GR (all this is covered in Schutz for instance). I only quote the ideal gas result because it is simple and because even at this level of approximation it is made clear that the kinetic energy of the particles in the fluid's rest frame does contribute. You can add viscosity or interparticle interactions or whatever you want, the kinetic energy still contributes.
Right, let's consider a simple scenario: Take a long rigid box - 10^6 m long, 1mx1m in crosss section, and as rigid as relativity allows (it's made of unobtanium). The walls of the box are very thin and lightweight - 1 gram per square meter, for a total box weight of 4000kg. The box is empty - almost! Inside is a single object - a 1kg unobtanium ball. This ball is bouncing from one end of the box to the other at some very high velocity. What is the gravitational mass of the box+ball system? Can this be proved without using general relativity? I think that the gravitational mass of the system is higher than the box+ball alone, and I think that this can be demonstrated using special relativity and newtonian gravity alone... I'll explain my reasoning later!
Aer: You have suggested no better alternative. In fact, you have suggested no alternative. I don't think you have the first clue about what you're talking about. That is incorrect. I am so disappointed in you.
What need is there for an alternative? I don't think you have the slightest clue as to what you are talking about.. No, it is not incorrect. When considering a single particle from a frame other than it's rest frame, it does not create any greater gravitational potential than if you considered the particle in its own rest frame.
Aer, I don't understand how you can have any further objection to the notion of kinetic energies of individual particles in the body's rest frame contributing to the gravitational field. I have directly quoted practicing GR physicists who say it does. You and I both know that the fact the authors were considering a perfect fluid has nothing to do with the conclusion that kinetic energy contributes. Whether or not we include interactions, viscosity, etc in our model of the bunch of particles, this not going to change the fact that kinetic energy in the rest frame contributes. Do you deny this?
You are new here, so you probably don't know how I operate. Anyway, that is what the following is for: I already know that physicists claim that it does, I am not arguing with that point. I specifically said on page 1 of this thread that "claims" are not proof and I will not accept them as such. I am not a student trying to understand General Relativity - I am past that. Simply put - I don't give a rat's ass about any claim any physicists makes without proof. Proof is experimental evidence. Now, as far as I know, there is no experiment to back up this claim and the questioning of the validity of the claim comes from the fact that it is inconsistent with the accepted fact that kinetic energy does not increase the gravitational energy of an individual particle such that the particle will never become a "black hole".
I'm sure you have a long record of peer-reviewed publications which establish your eminence in this field.
No it's not. If you have one particle in a single box, it is no different. For the case of the box, you say the kinetic energy contributes. For the case of just the particle by itself, the kinetic energy does not contribute. To borrow from MacM (whom I disagree with, but anyway...) don't quote the good book Please Register or Log in to view the hidden image!
I didn't claim to be a physicist. I just said I don't want to be spoon fed information to get "grades". I want the proof, the same proof you should want - but perhaps you are incompetent Please Register or Log in to view the hidden image!
Aer: If only you had something to support your claim. You complain about me quoting "the book" (i.e. established, accepted physics). But it's just empty rebellion for its own sake.
If only you had something to support the claim made by the physicists you quote. Call it rebellion all you want. I provided you claims by other physicists that kinetic energy does not contribute to the gravitation that a single particle will create (i.e. the black hole issue).
