- #1
Terilien
- 140
- 0
why do we define it that way? What properties make it the best possible choice for the gravitational field?
Question about the energy momentum tensor
In summary: It just seems like there should be some other way that gravity and the other fields could interact. But when you think about it, it makes sense. The action of gravity is just a vector addition of the actions of the other fields.
- #2
- 10,338
- 1,516
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.
- #3
smallphi
- 441
- 2
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
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
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- #4
pmb_phy
- 2,952
- 1
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.Terilien said:
Pete
- #5
- 6,993
- 2,470
smallphi said:
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".
- #6
MeJennifer
- 2,008
- 6
robphy said:
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?
- #7
smallphi
- 441
- 2
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
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
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- #8
MeJennifer
- 2,008
- 6
So are you saying that electro-magnetism does not contribute to the stress-energy tensor?smallphi said:
- #9
smallphi
- 441
- 2
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.
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.
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FAQ: Question about the energy momentum tensor
What is the energy momentum tensor?
The energy momentum tensor is a mathematical object used in the field of physics to describe the distribution of energy and momentum in space and time. It is a rank-2 tensor that contains 10 components, representing the energy density, momentum density, and stress components in three dimensions.
How is the energy momentum tensor used in physics?
The energy momentum tensor is a key tool in Einstein's theory of general relativity and is used to describe the curvature of spacetime due to the presence of energy and momentum. It is also used in other areas of physics such as fluid dynamics and cosmology to analyze the flow of energy and momentum.
How is the energy momentum tensor calculated?
The energy momentum tensor can be calculated using the equations of motion for a given physical system. In general relativity, it is derived from the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. In other areas of physics, it may be calculated using other equations and principles specific to the system being studied.
What are the physical implications of the energy momentum tensor?
The energy momentum tensor provides important insights into the behavior of matter and energy in the universe. It helps us understand the distribution of energy and momentum in space and time and how they interact with each other and with the curvature of spacetime. It also allows us to make predictions and calculations about the behavior of physical systems.
Can the energy momentum tensor be directly observed?
No, the energy momentum tensor itself cannot be directly observed. It is a mathematical construct used to describe physical phenomena and is not directly measurable. However, its effects can be observed and measured through other physical quantities such as gravitational fields, energy flux, and momentum transfer.