Question about the energy momentum tensor

[Terilien]

why do we define it that way? What properties make it the best possible choice for the gravitational field?

 

[pervect]

The properties that make the energy momentum tensor good for relativity is that it is a tensor, i.e. it transforms in the standard way that tensors transform with boosts, etc. The energy momentum 4-vector also has this property but only for point particles (particles of zero volume) or for isolated systems. The energy-momentum 4-vector of a non-isolated system (i.e. a piece of a larger system) is not, in general covariant. (I can give a reference if one is needed). This is why one needs the stress-energy tensor.

 

[smallphi]

Even classical physics uses a stress tensor to describe how a different fluid elements of an extended body interact with each other (through pressure, viscosity etc.).

The relativistic energy momentum tensor is the generalization of the classical one. You need a tensor field, whose components in general vary from one spacetime point to another, to describe an extended body (fluid) in GR. Only one energy momentum 4-vector field simply doesn't contain enough information how different parts of the fluid interact.

In GR, gravity is described as curvature of the manifold, all other forces are captured by the energy momentum tensor.

Here is mor info:
http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html

 

[pmb_phy]

why do we define it that way? What properties make it the best possible choice for the gravitational field?

Einstein gave this as the reason for using this tensor - The stress-energy-momentum tensor gives the correct and complete description of mass and since mass is equivalent to energy if then follows that since we know that mass is the source of gravity it then follows that this tensor should be the source.

Pete

 

[robphy]

In GR, gravity is described as curvature of the manifold, all other forces are captured by the energy momentum tensor.

I think it is more correct to say that
"all [non-gravitational] 'matter fields' contribute to the gravitational field via the energy momentum tensor [via the Einstein Equations]",
however,
"the other forces (like electromagnetism) are captured by other field-equations (like the Maxwell Equations) that those fields satisfy".

 

[MeJennifer]

I think it is more correct to say that
"all [non-gravitational] 'matter fields' contribute to the gravitational field via the energy momentum tensor [via the Einstein Equations]",
however,
"the other forces (like electromagnetism) are captured by other field-equations (like the Maxwell Equations) that those fields satisfy".

As far as I know the EM tensor covers:

- The density of energy, including rest mass and the energy of electro-magnetism
- The flux of this energy
- The density of momentum
- The flux of this momentum

Am I wrong?

 

[smallphi]

Yes. Both Einstein equations(which are equations of motion for the metric) and equations of motion for other fields are produced from the action. The total action of the system

S = Sgravity + Sother_fields,

depends on the metric and the other fields as free variables.

Setting to zero the variations of S with respect to the metric produces the Einstein equations which show how the derivatives of the metric (the curvature) respond to the energy momentum tensor of the other fields. Hence the energy momentum tensor of the other fields captures their gravitational effects.

Setting to zero the variations of S with respect to other fields, produces their equations of motion which capture their dynamics. Carroll section 4.3

 

[MeJennifer]

Yes

So are you saying that electro-magnetism does not contribute to the stress-energy tensor?

 

[smallphi]

The 'Yes' was to Robphy not to you. I didn't refresh the page so I didn't notice another post after him. Of course the EM tensor contributes to the energymom. tensor of the fields.

Can someone explain or at least motivate why the actions of gravity and the other fields simply add in the total action instead of composing them in a more complicated ways:

S = Sgravity + Sother_fields

When I first saw it, I was shocked the combination is that simple.