Gravitational Pseudotensor Requirements

[Mueiz]

Why is it always stated that one of the requirements of the Gravitational Stress-Momentum Pseudotensor that it vanish locally in an inertial frame? In GR either there is a gravitational field or there is none according the Riemann tensor which is absolute.

 

[jambaugh]

Why is it always stated that one of the requirements of the Gravitational Stress-Momentum Pseudotensor that it vanish locally in an inertial frame? In GR either there is a gravitational field or there is none according the Riemann tensor which is absolute.

Check your source, is it not the affine connection which vanishes locally in an inertial frame?

 

[Mueiz]

Check your source, is it not the affine connection which vanishes locally in an inertial frame?

What vanishes locally in an inertial frame is the Gravitational Stress-Momentum Pseudotensor as a requirement ...so am asking about the physical postulate behind this requirement

 

[Bill_K]

The Riemann tensor describes the presence of a gravitational field. The stress-energy pseudotensor describes something else. You derive it by going to a locally Galilean coordinate system where T^μν,v = 0. Then you relax the coordinate condition and see that the divergenceless quantity is now T^μν plus something else, which you call t^μν. In fact t^μν is algebraically related (no derivatives) to the affine connection, so it vanishes when the affine connection vanishes.

 

[Mueiz]

The Riemann tensor describes the presence of a gravitational field. The stress-energy pseudotensor describes something else. You derive it by going to a locally Galilean coordinate system where T^μν,v = 0. Then you relax the coordinate condition and see that the divergenceless quantity is now T^μν plus something else, which you call t^μν. In fact t^μν is algebraically related (no derivatives) to the affine connection, so it vanishes when the affine connection vanishes.

Thanks ! that is true.. but this is not an answer to my question .
My question is very specific :what is the physical postulate behind the requirement that the Gravitational Stress-Momentum Pseudotensor must vanish locally in an inertial frame
I hope I will get a specific answer so that we do not crowd the thread with well known information
I mentioned in my first post that in GR either there is a gravitational field or there is none according the Riemann tensor which is absolute only to confirm the fact that the absence of gravitational field can not be the reason.

 

[Bill_K]

Mueiz, I answered your specific question, and agree that if you still have trouble seeing it, further discussion would be redundant. The pseudotensor does not represent the presence of a gravitational field. Its purpose is to obtain a quantity whose coordinate divergence vanishes and thus leads to a conservation law. Perhaps that strikes you as unphysical, but that is the motivation behind it.

 

[Mueiz]

You speak about the vanishing of the coordinate divergence which represent the requirement of the law of conservation but I speak about the local vanishing of the Gravitational Stress-Momentum Pseudotensor itself .

 

[cosmik debris]

A simple answer maybe: so that a locally inertial frame has a Lorentz metric.

 

[jambaugh]

Thanks ! that is true.. but this is not an answer to my question .
My question is very specific :what is the physical postulate behind the requirement that the Gravitational Stress-Momentum Pseudotensor must vanish locally in an inertial frame
I hope I will get a specific answer so that we do not crowd the thread with well known information
I mentioned in my first post that in GR either there is a gravitational field or there is none according the Riemann tensor which is absolute only to confirm the fact that the absence of gravitational field can not be the reason.

You're insisting on a physical meaning to a not-quite-physical correction term. Since the G S-M PseudoT is the correction to the linearized Einstein tensor added to yield the full Einstein tensor, it's vanishing just goes to show that in a locally inertial frame you already have a (locally) linearized field. That's just the definition of a locally inertial frame its a tangent (cotangent?) linear frame.

 

[Mueiz]

You're insisting on a physical meaning to a not-quite-physical correction term. Since the G S-M PseudoT is the correction to the linearized Einstein tensor added to yield the full Einstein tensor, it's vanishing just goes to show that in a locally inertial frame you already have a (locally) linearized field. That's just the definition of a locally inertial frame its a tangent (cotangent?) linear frame.

I wonder that if we have smooth curved space-time (looks like your beautiful hat) one expect that locally inertial frame has a Lorentz metric but why should he assume locally vanishing of G S-M PseudoT .. is it not true that the existence of a density of energy is the sufficient and necessary condition for the field (which exist regardless of the reference frame)?
Or does a Lorentz metric means zero G S-M PseudoT?!

 

[QZQZ]

I believe that the gravitational energy pseudotensor is intended to measure the amount of energy carried by gravitational waves at a point. Relative to a freely falling rest frame, there is no energy in those gravitational waves.