What is the purpose of the Einstein stress-energy tensor?

In summary, the stress-energy tensor is a coordinate-free representation of mass-energy and momentum in Einstein's gravity equation. It is a function that takes two vectors as inputs and gives a real scalar as output, and can be described in multiple versions, including the traditional matrix representation.
  • #1
AleksanderPhy
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Hello I'm new here on this forum and on physics too.
I have problem on Einstein famous equation
einstein.png

I have a problem on the last component Tαβ I know that tensor name is Einstein stress-energy tensor and I know that Tαβ is:https://qph.is.quoracdn.net/main-qimg-c810f8bca07c6e580138cc1906a693bf?convert_to_webp=true [Broken] but can we describe it some other versions not like matrics version
 
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  • #2
The stress-energy tensor is a way of representing the more familiar notions of mass-energy and momentum in a way that is independent of reference frame - ie the coordinate system used. It's easy enough to see that traditional measurements of momentum are coordinate dependent - they vary between reference frames. It is less obvious, but nevertheless true, that measurements of mass-energy also depend on reference frame.

Einstein's gravity equation is designed to express the dependence of the curvature of spacetime on some measure of mass-energy. But since the curvature is coordinate-free, the measure of mass-energy needs to be coordinate-free too. The stress-energy tensor is that coordinate-free 'object'.

The matrix you have written above is a representation of the tensor in coordinates. But note it is the representation that uses coordinates, and is coordinate-dependent, not the tensor itself. That is, the 4 x 4 matrix is not the tensor but a representation of it. Although one sometimes relaxes one's precision and refers to the matrix as a tensor.

Mathematically, the tensor is a function that takes two vectors as inputs and gives a real scalar as output. You can if you wish think of the first vector as answering the question 'what do you want to measure?' (to which the answer will be something like 'energy, x-momentum, y-momentum or z-momentum' or a combination thereof) and the second vector as answering the question 'across what surface do you want to measure the flow of the quantity of the first vector?' (the second vector is the normal to the surface)
 

What is the Einstein stress-energy tensor?

The Einstein stress-energy tensor is a mathematical tool used in the theory of general relativity to describe the distribution of energy and momentum in a given space-time. It is a symmetric tensor that relates the curvature of space-time to the energy and momentum present in that space-time.

How is the Einstein stress-energy tensor calculated?

The Einstein stress-energy tensor is calculated using the Einstein field equations, which relate the curvature of space-time to the distribution of energy and momentum. It is calculated by taking the second derivative of the metric tensor, which describes the curvature of space-time, and subtracting a term involving the energy-momentum tensor, which describes the distribution of energy and momentum.

What is the significance of the Einstein stress-energy tensor?

The Einstein stress-energy tensor is significant because it is a crucial component of the theory of general relativity and allows us to understand the relationship between energy, momentum, and the curvature of space-time. It is also used in many practical applications, such as predicting the behavior of massive objects in space and understanding the evolution of the universe.

What are the units of the Einstein stress-energy tensor?

The units of the Einstein stress-energy tensor are energy density per unit volume for the diagonal components and energy flux per unit area for the off-diagonal components. In SI units, the stress-energy tensor is typically measured in joules per cubic meter for energy density and joules per square meter for energy flux.

How does the Einstein stress-energy tensor relate to the conservation of energy and momentum?

The Einstein stress-energy tensor is a manifestation of the principle of covariant conservation, which states that the sum of the energy and momentum within a given space-time is conserved. This means that the Einstein stress-energy tensor must satisfy certain mathematical properties, ensuring that energy and momentum are conserved in any physical process described by the theory of general relativity.

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