# The energy-momentum tensor and the equivalence principle

• hellfire
In summary, the only way to have a theory of gravitation that fulfills the equivalence principle is to make use of a tensor as the source of gravity, such as the Einstein tensor Gab. This is supported by the principle of general covariance, which requires the laws of nature to be the same in all coordinate systems. While some theories may introduce a scalar field, it must still be coupled to a tensor in order to maintain covariance. The Brans Dicke theory is an example of a theory that includes a scalar field, but the field equation is still of second rank and requires a tensor component.
hellfire
Is it correct that the only way to have a theory of gravitation that fulfills the equivalence principle is to make use of a tensor as the source of gravity (and not a scalar or a vector, for example)? How can this be proven?

If gravity is to be described by the Einstein tensor Gab then its source must be a tensor as well.

Most tests of GR to date are in vacuo, that is all they are testing is
Gab = 0 ; it seems to work pretty well!

Garth

Dirac's short book gives a nice insight to this subject:couplings of gravity and matter.No wonder,Dirac was a field theorist,like Pauli,Feynman,Weinberg...

Daniel.

hellfire said:
Is it correct that the only way to have a theory of gravitation that fulfills the equivalence principle is to make use of a tensor as the source of gravity (and not a scalar or a vector, for example)? How can this be proven?
A scalar and a vector are both tensors. From what I recall there is a theory by Dicke (Brans too?) which is a relativistic theory of gravity which is consistent with the equivalence principle.

Pete

Yes, the Brans Dicke theory takes the Einstein field equation and adds a scalar field coupled to the (rest) mass density of the universe that endows particles with inertial mass. It retains the equivalence principle and so the effect of this scalar field is to vary the 'Gravitational 'constant''. Although many attempts to integrate QT with GR would like such a scalar field the BD theory is not verified observationally in solar system experiments; the scalar field perturbs space-time.

For information SCC modifies this theory by allowing particle masses to vary (with gravitational potential energy) and G then becomes observationally constant. It breaks the equivalence principle but nevertheless is consistent with solar system tests.

Garth

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As far as I know, scalar (rank 0 tensor) theories of gravity in which the source of gravity is the trace of the energy-momentum tensor do not fulfill the equivalence principle, since light does not couple to gravity (the trace of the electromagnetic energy-momentum tensor is zero). The question is whether a vector (rank 1 tensor) theory of gravity may fulfill the equivalence principle or whether only theories in which the source of gravity is at least a rank 2 tensor do.

There is no way to avoid a tensor description of gravity geometrically.

Chronos said:
There is no way to avoid a tensor description of gravity geometrically.
That's the kind of answer I am looking for… but with a proof.

hellfire said:
That's the kind of answer I am looking for… but with a proof.
You're incorrectly expecting proof where there is only postulate. The principle of general covariance requires the laws of nature to be the same in all coordinate systems. We give the name "tensor" to those objects which satisfy this property of covariance.

Pete

pmb_phy said:
You're incorrectly expecting proof where there is only postulate. The principle of general covariance requires the laws of nature to be the same in all coordinate systems. We give the name "tensor" to those objects which satisfy this property of covariance.

Pete
But a theory in which gravity couples to the trace of the energy-momentum tensor would be also covariant, as the trace is scalar (a rank 0 tensor). But it does not fulfill the equivalence principle.

hellfire said:
But a theory in which gravity couples to the trace of the energy-momentum tensor would be also covariant, as the trace is scalar (a rank 0 tensor). But it does not fulfill the equivalence principle.
Since when?

Pete

pmb_phy said:
Since when?

Pete
I am not aware of any error in what I wrote, but If I wrote something wrong, please correct me. That's the best way for me to learn.

What Pete said - GR treats spacetime as a four dimensional manifold. To describe the geometry of such a manifold without introducing frame dependence, you must use a rank 2 tensor.

Chronos said:
What Pete said - GR treats spacetime as a four dimensional manifold. To describe the geometry of such a manifold without introducing frame dependence, you must use a rank 2 tensor.
That is not true. The Branse Dicke theory treats spacetime as a 4-d manifold and the description of the geometry is not frame dependant. However the field equations need not be a second rank tensor (e.g. Brans Dicke).

Pete

Hmmm... The Brans Dicke theory has two separate field equations and an equation of state, The gravitational field equation certainly is an equation of second rank tensors, the scalar field equation is an equation in which each term is a scalar, as is the equation of state, however, that is no different to GR, which also requires a 'scalar' equation of state. [Though no scalar field equation]

Garth

Garth said:
Hmmm... The Brans Dicke theory has two separate field equations and an equation of state, The gravitational field equation certainly is an equation of second rank tensors, the scalar field equation is an equation in which each term is a scalar, as is the equation of state, however, that is no different to GR, which also requires a 'scalar' equation of state. [Though no scalar field equation]

Garth
What are these two separate field equations you speak of??

Note: I made an error above. The trace can't be the source of gravity because for a beam of directed light the trace is zero and since mass is equivalent to energy the trace can't be a source.

I believe that in the Brans Dicke theory there is a scalar field but the field equation is second rank. The d'Lanbertian of the scalar field is proportional to the energy momentum tensor in that theory.

Is that correct pervect? You seem to know more about it than I do.

Pete

I definitely know it's Jean le Rond d'Alembert and,consequently,the operator's name is "d'Alembertian".

Daniel.

Garth said:
Yes, the Brans Dicke theory takes the Einstein field equation and adds a scalar field coupled to the (rest) mass density of the universe that endows particles with inertial mass. Garth

Is that scalar, in the equation or the Lagrangian, multiplied by something (metric tensor?) to bring it up to second rank and make the whole thing homogeneous? As you know that is how the cosmological constant is brought into the field equation in GR.

Pete - As I explained - (although obviously rather obscurely!) The two BD field equations are:
1. The gravitational field equation; Einstein's with G replaced by Phi-1 and the stress-energy-momentum tensor of the scalar field TPhi ab added to that of normal matter-energy. The result is a homogeneous second rank tensor equation.

2. The scalar field equation; The d'Alembertian of Phi coupled to the trace of the matter stress-energy-momentum tensor. In this equation each term is a scalar and therefore selfAdjoint homogeneous.

Garth

I am sorry but I still don’t get it, so please be patient with me.

A theory in which gravity would couple only to the trace of the energy-momentum tensor (such a theory was considered by Einstein before general relativity):

$$\square \phi = - 4 \pi G T_{\mu}^{\mu}$$

With $$\inline g_{\mu \nu}$$ being diagonal with $$\inline \phi$$ or $$\inline - \phi$$ as diagonal elements...

a)...is a covariant theory, isn’t it?

b) but, however, it does not satisfy the equivalence principle, since light would not couple to a gravitational field. A photon would not be redshifted or blueshifted in a gravitational field, but it would be redshifted or blueshifted when emitted from an accelerated frame.

If all above is correct, what are the reasons, if any, for the need of having a rank 2 tensor in order to satisfy the equivalence principle?

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hellfire said:
a)...is a covariant theory, isn’t it?

Yes. As a side note, almost any theory can be put into covariant form. Whether it looks nice when written that way is another question.

b) but, however, it does not satisfy the equivalence principle, since light would not couple to a gravitational field. A photon would not be redshifted or blueshifted in a gravitational field, but it would be redshifted or blueshifted when emitted from an accelerated frame.

Before saying that, you have to prescribe how matter couples to the field. In GR, this is automatic (due to the Bianchi identities). Here, you have to introduce additional postulates.

I'll assume for simplicity that you want light to move along null geodesics. If so, then the coupling of $$\phi$$ to $$g_{\mu \nu}$$ automatically implies that light is affected by gravity. The only difference is that electromagnetic field is not a (direct) source for the gravitational one. I'm sure there would be different predictions for light bending etc., but I don't want to work them out.

Also, to nitpick a little, the metric should have components like $$1+ \phi$$, not $$\phi$$ all by itself. Otherwise, spacetime wouldn't be well-defined

If all above is correct, what are the reasons, if any, for the need of having a rank 2 tensor in order to satisfy the equivalence principle?

There aren't any. The statements that scalar and vector gravity don't work are really saying that specific theories with simple Lagrangians have been found to disagree with experiment. That doesn't mean that you couldn't make up another Lagrangian that would pass all the tests.

Also, any theory can be written in an infinite number of different ways. You shouldn't put too much emphasis on the most popular one. For example, Ashtekar's formulation of GR looks completely different from Einstein's/Hilbert's, but it's the same thing.

Even if you wanted to stay with something of the form $$G_{\mu \nu} = (\ldots)$$, you can still come up with plenty of source terms other than the (standard) stress-energy tensor.

Thank you Stingray. I have the impression I was mixing two things. The field equation which relates the gravitational field to the energy-momentum tensor makes only a statement about how the gravitational field is produced by matter, but makes no statement about the way how matter or other fields behave inside a gravitational field. For that, additional conditions are necessary (i.e. the Lagrangian for the matter fields has to be defined). This second aspect is relevant for the equivalence principle, but not the first.

hellfire said:
I am sorry but I still don’t get it, so please be patient with me.

A theory in which gravity would couple only to the trace of the energy-momentum tensor (such a theory was considered by Einstein before general relativity):

$$\square \phi = - 4 \pi G T_{\mu}^{\mu}$$

With $$\inline g_{\mu \nu}$$ being diagonal with $$\inline \phi$$ or $$\inline - \phi$$ as diagonal elements...

a)...is a covariant theory, isn’t it?

b) but, however, it does not satisfy the equivalence principle, since light would not couple to a gravitational field. A photon would not be redshifted or blueshifted in a gravitational field, but it would be redshifted or blueshifted when emitted from an accelerated frame.

If all above is correct, what are the reasons, if any, for the need of having a rank 2 tensor in order to satisfy the equivalence principle?
Also Black Holes would not be coupled to a gravitational field. As the matter of a collapsed star became degenerate and relativistic its gravitational field would disappear!
Garth

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Thanks for the converasation folks. For the most part I think I'll have to nod off on the Brans-Dicke stuff at this point. At least for now. I've never rigorously learned it up to this point and as I've read it in Wienberg the last few days I've made mistakes in the reading. That's not all that uncommon for the average gent but it appears that the old peepers are now unable to read a book without glasses.

Please note that there are two different symbols used to express the D'Alambertion operator. One is a little box and the other is the square of a little box.

Pete

## 1. What is the energy-momentum tensor?

The energy-momentum tensor is a mathematical quantity in physics that describes the distribution of energy and momentum in a given system. It is represented as a matrix with 4 rows and 4 columns, where each element represents a different component of energy or momentum.

## 2. How is the energy-momentum tensor related to the equivalence principle?

The equivalence principle states that the effects of gravity are indistinguishable from the effects of acceleration. The energy-momentum tensor is used to describe the effects of gravity in Einstein's theory of general relativity, making it a fundamental concept in understanding the equivalence principle.

## 3. What is the significance of the energy-momentum tensor in physics?

The energy-momentum tensor is significant because it allows us to accurately calculate and predict the behavior of systems in the presence of gravity. It is a crucial component in Einstein's theory of general relativity, which is the most accurate theory of gravity we have to date.

## 4. How is the energy-momentum tensor calculated?

The energy-momentum tensor is calculated using mathematical equations that take into account the distribution of energy and momentum in a given system. These equations involve concepts such as spacetime curvature and the energy-momentum density of matter and energy sources.

## 5. Can the energy-momentum tensor be applied to all systems?

The energy-momentum tensor can be applied to any system that involves energy and momentum, including both classical and quantum systems. However, it is most commonly used in the context of general relativity and gravitational systems.

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