Since it appears you do not even acknowledge the situation in classical mechanics, I feel there is no common ground for us to discuss GR. It would just be a waste of time. Besides, we have already been through all of this on another thread. It seems you have chosen to substitute the established definition of the energy-stress tensor with your own strange ideas; while you are free to do so, that sadly implies you will have no choice but to reject GR as a valid theory. I wish you good luck with that.
Opinions are no substitute for facts Markus. Repeating from last post: Well - can you? If not, where does that leave your (and one or two others) notion of stress is outright an energy density?
What I'm finding curious about the discussion so far, is that I thought what pressure is, and how it's related to energy density (for instance, P = F/A is equivalent to Joules per cubic metre--the units match), was established on page 2. Maybe we could discuss the difference between pressure in three spatial dimensions and in one time dimension. Why is pressure a "momentum product'', or rather is the flow of momentum? I think it's important to see the connections between momentum, energy, and pressure.
Correct me if I'm wrong, but area has units of m^2. Area is two dimensional, whereas volume is three dimensional and has units of m^3. http://en.wikipedia.org/wiki/Area http://en.wikipedia.org/wiki/Volume
*sigh* I don't know why I even bother. No matter what I present here now, you will reject it anyway. Still, here is the general outline - for a solid body with induced stresses we decompose the SEM into a kinetic part and an elastic stress part : \(\displaystyle{T^{\mu \nu }=K^{\mu \nu }+E^{\mu \nu }}\) The elastic stress tensor E can, in the nonrelativistic limit, be shown ( http://www-personal.umich.edu/~williams/papers/elastensor.pdf ) to be a function of the classic stress tensor \(\sigma\) like so : \(\displaystyle{E^{00}=\sigma ^{00}}\) \(\displaystyle{E^{0j}=0}\) \(\displaystyle{E^{i0}=v^{i}\sigma ^{00}+E^{ij}v_{j}}\) \(\displaystyle{E^{ij}=\frac{\partial x^i}{\partial \alpha ^{s}}\frac{\partial x^j}{\partial \alpha ^t}\sigma ^{st}}\) This provides the connection between classic stress tensor \(\sigma\) and relativistic elastic stress tensor E, in the non-relativistic limit. As for your question regardling stored potential energy, the answer is then \(\displaystyle{W=\frac{1}{2}\sum_{i}\sum_{j}\sigma _{ij}\varepsilon _{ij}}\) wherein \(\varepsilon\) is the classic strain tensor, defined in the usual way in terms of infinitesimal replacements. This shows how your potential stored energy follows from the classic stress and strain tensors, and hence by extension from the SEM tensor. Note that the expression you gave in post 54 is applicable only to the special case of isotropic media, whereas the above is the more general expression for any type of medium. This leaves us exactly in the realm of pretty much any textbook on classical mechanics which you care to reference, and it also demonstrates my earlier point that the relativistic SEM tensor is really just a covariant generalization of its classic mechanical counterpart. If you need more references, you might also look at http://www.mathem.pub.ro/apps/v13/A13-ci.pdf
Well, correct me if I'm wrong, but force has units of mass x acceleration. That's kg.m.s[sup]-2[/sup]. Divide that by m[sup]2[/sup] for some area, you have: kg.m[sup]-1[/sup].s[sup]-2[/sup]. You get the same thing dividing energy in Joules by m[sup]3[/sup]. Two ways to get the same thing, so pressure and energy density are equivalent in units. Does that mean they're equivalent 'physically', or does it mean something else?
First off, do you agree with my post and see where your error is? Next. I'll correct you, because you are wrong: http://en.wikipedia.org/wiki/Force
Correct, I am saying there is force (as I quoted from Wiki) and then there is acceleration. Two different animals. F=ma is a NET FORCE. So do you now see where all your errors are?
No I'm still a bit lost. The quote from Wiki also says force has units of Newtons, so these Newtons don't have the same units as mass x acceleration?
And see this ( http://en.wikipedia.org/wiki/Pressure ) : "Since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume measured in J·m−3, related to energy density."
Or this ( http://en.wikipedia.org/wiki/Energy_density ): "Energy per unit volume has the same physical units as pressure, and in many circumstances is a synonym"
I've definitely lost interest in whatever it is you think is a point. A force has units of mass x acceleration, so does a Newton. You disagree with me, but also with lots of other people who understand forces, acceleration and what force per unit area means. You're on your own with whatever it is you think is "wrong" because there is nothing wrong.
Well, I've explained it, given quotes from Wiki, showed you your errors, and corrected them all. You can lead a horse to water.......so I guess I'm done here. Carry on.
None of those links work, but no matter. You give the expressions for general stress and strain and elastic energy density. Fine. But fail to mention one crucial thing. That W for elastic energy density belongs where in the SET? I suggest solely in the Too term as earlier claimed. You have maintained all along stress/pressure IS a form of energy density. Hence an 'energy density' directly proportional to stress, quite unlike the quadratic relationship applying to elastic energy density (as I have very oft repeated elsewhere). Then it must make an appearance additional to elastic energy density. So I again ask - show where in that yes simple isotropic case in #54, where this extra stress-on-it's-own energy density is. An 'energy density' proportional directly to stress (that's how it acts as a source term in GR after all). Quantify it. Then perform the impossible task of balancing the energy ledger - given that input energy has, by definition of how it's calculated, been already exactly accounted for in the stored elastic energy density I gave. Good luck. As for your #72-#74, nothing in those references given contradicts my position, in particular they do not declare stress/pressure IS energy density, as a careful reading in context reveals.
They work just fine for me. To be honest, I have no idea what you are on about here, and your expression "energy density directly proportional to stress" is physically meaningless. Stress is energy. There is no distinction. Your example is a solid put under uniaxial stress, so stress energy is the only energy present. If you wish to consider pressure in the material as well then you are free to do so, and other terms will appear in your sum. This is why a summation appears in the formula I provided. Where's the problem ?? The relation I gave you is the total value of the energy stored, and it is the summation over all the indices of the tensors, giving a linear combination of terms, each of which has the dimension of energy. There is nothing "missing" and nothing "to be balanced" here; the different components of the stress tensor don't have different units - they are all just energy, so that the sum of them is energy too. There are no issues at all here, unless of course you deny that the components of these tensors are indeed energies, but then you are just inventing difficulties where there are none. Frankly speaking, that is your own problem. The definition of the SEM tensor in GR is clear and unambiguous, and can be found in any textbook on the subject matter ( I won't bother with any more references since, as proven below, you will just reject them anyway ). Your refusal to accept those definitions changes not the fact that the results obtained from GR using that SEM tensor are in good accordance with experiment and observation. Ultimately, that is what we measure the success or failure of the theory against. A quick search of the forum reveals that you have been proliferating your position for some time on a number of threads, and also that anyone with any real understanding of GR has yet to agree with you that stress and pressure are not forms of energy. To me, that is not very surprising. You see, this is why I said we have no common ground to discuss this issue; you simply aren't prepared to even consider the possibility that you might be wrong, and you don't acknowledge or outright reject any references given which runs contrary to your belief. I'll leave you to it. The interested reader can make up his/her own mind given the four references I provided. I don't really have anything else to add to them.
