# Details of Stress-Energy-Momentum Tensor

• TheEtherWind
In summary, there are some questions about the stress-energy-momentum and its components. The stress-energy-momentum tensor is symmetric and the wikipedia article shows energy flux across the top row vector and momentum density down the first column vector. The 0,0 component can be either the energy density or the relativistic mass density, depending on the convention used. The energy density is the sum of mechanical and electromagnetic energy, while the momentum density and stresses are also a combination of these energies. However, the energy flux and momentum density are equivalent concepts. Tensors assume c=1, making mass and energy equivalent, and the flux of mass and density of momentum are ordinary while the flux of energy would be much larger.
TheEtherWind
I'm fairly familiar with the elements of the stress-energy-momentum, but I have some questions. Is it right to say the stress-energy-momentum tensor is symmetric? And if so, why does the wikipedia article show energy flux across the top row vector, and momentum density down the first column vector? Or am I messing up the meaning of 'energy flux?' Maybe they're the same?

As far as the 0,0 component goes.. I was watching this video lecture from MIT. You can find it by searching "Lec 3 | MIT 8.224 Exploring Black Holes" on youtube. And he uses the energy density as the 0,0 component. However, the wikipedia article states the 0,0 component is the relativistic mass density. I understand they only differ by a factor of c2 but which is correct to use?

Also, the element of 'energy density,' is it simply the addition of the mechanical energy and the electromagnetic energy? And the same question goes for the momentum density, and stresses. Do you simply add the mechanical and electromagnetic contributions?

These descriptions are all true, so that a logical consequence of them is that the energy flux happens to equal the momentum density (*c2), while these are 2 different concepts.
Usual conventions on tensors assume c=1 so that "mass" and energy are the same. If we don't put c=1, as the stress-energy tensor is defined as twice contravariant, its 0,0 component is the mass density; its 0,i and i,0 equal components are the density of momentum and the flux of "mass" where "mass" = energy * c2.
You can see this considering that for a twice contravariant tensor, the space and time components have the same magnitude when describing an object going at 1 m/s, as they are given by the tensor product of both parallel vectors (1,speed) and (mass, momentum) that are tangent to the movement of the object in space-time. The magnitudes of the flux of mass and density of momentum are ordinary, while the flux of energy would be huge at is is the energy of mass mc2 of the object that is moved.

(I have an introduction to tensors in my site : settheory.net)

The energy, is the sum of all possible forms of energies for all kinds of particles and forces. (We may argue that the "mechanical energy", beyond the energy of mass, is but a hidden form of electromagnetic energy)

## 1. What is the Stress-Energy-Momentum Tensor?

The Stress-Energy-Momentum Tensor is a mathematical object used in Einstein's theory of general relativity to describe the distribution of energy, momentum, and stress in a given region of space. It is a 4x4 matrix that represents the energy-momentum density and flux of a system at a specific point in spacetime.

## 2. How is the Stress-Energy-Momentum Tensor calculated?

The Stress-Energy-Momentum Tensor is calculated using the energy-momentum density and flux of a system. This includes the mass density, energy density, and momentum density of all forms of matter and energy present. The tensor is then constructed using the Einstein field equations and the curvature of spacetime.

## 3. What is the significance of the Stress-Energy-Momentum Tensor?

The Stress-Energy-Momentum Tensor is significant because it allows us to understand how matter and energy interact with the fabric of spacetime. It is essential in Einstein's theory of general relativity and is used to describe the behavior of gravity and the curvature of spacetime.

## 4. Can the Stress-Energy-Momentum Tensor be used to predict the behavior of matter and energy?

Yes, the Stress-Energy-Momentum Tensor is a powerful tool in predicting the behavior of matter and energy. By understanding the distribution of energy and momentum in a given system, we can make predictions about how that system will behave in different conditions.

## 5. How does the Stress-Energy-Momentum Tensor relate to the conservation laws of physics?

The Stress-Energy-Momentum Tensor is closely related to the conservation laws of physics, such as the conservation of energy and momentum. This is because the tensor represents the total energy and momentum present in a system, and these quantities are conserved throughout time, according to the laws of physics.

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