View Full Version : Stress Energy Tensor

tommac

2010-Jan-21, 06:33 PM

I am trying to figure out in laymans terms what the stress energy and metric tensors mean.

Is the stress energy tensor the net of all energy that affects a certain point? I guess this would need to include the distance and direction of each mass of a system?

Then the metric tensor is the amount of space time curvature that is the result of the that energy?

Do I understand this correctly ( at least at this level ) ?

Tensor

2010-Jan-21, 10:04 PM

I am trying to figure out in laymans terms what the stress energy and metric tensors mean.

Is the stress energy tensor the net of all energy that affects a certain point? I guess this would need to include the distance and direction of each mass of a system?

Then the metric tensor is the amount of space time curvature that is the result of the that energy?

Do I understand this correctly ( at least at this level ) ?

Not quite. The stress-energy tensor includes the energy density (or the mass times c2, the momentum in each of the spatial directions, the flux of momentum (the amount of momentum through a given area during a given time), the flux of energy, and the pressure. Pressure doesn't really amount to much until you get to neutron star densities and pressures.

There are sixteen terms in the stress-energy tensor. Each of those terms is not just a value, but an equation (and actually, because of symmetry, only ten are needed, the others cancel). Each of those equations is a hyperbolic, partial differential equation.

The metric tensor tells you how to measure distances on the manifold. Distances can be measured differently depending on the curvature in each direction (including the time dimension).

The amount of space-time curvature is given by the Ricci Curvature Tensor. If you want to see what the equations in the Ricci Tensor look like, try [/url=http://archive.ncsa.illinois.edu/Cyberia/NumRel/mathmine1.html]this [/url] (remember, there are two more pages of equations, click on want to see more).

tommac

2010-Jan-22, 09:10 PM

Not quite. The stress-energy tensor includes the energy density (or the mass times c2, the momentum in each of the spatial directions, the flux of momentum (the amount of momentum through a given area during a given time), the flux of energy, and the pressure. Pressure doesn't really amount to much until you get to neutron star densities and pressures. .

But at the end of the day this allows one to calculate a value at a certain point right? Like can I use the stress energy tensor to calculate the sum of all those things you mention at the tip of my finger at a certain instant of time? AND when you sum up all of that stuff ... what are we really talking about? We are talking about the effects of a different forms of energy at a very specific point in time and space? Isnt the net of (energy density (or the mass times c2, the momentum in each of the spatial directions, the flux of momentum (the amount of momentum through a given area during a given time), the flux of energy, and the pressure. ) really just energy in the end?

publius

2010-Jan-22, 11:30 PM

There's a rather cute little way think of the components of the stress-energy tensor. The lower 3x3 part, the spatial (or space-space) parts are just the familiar 3D stress (momentum flux) tensor. That is the component T_ij is the flux of the ith component of momentum in the jth direction.

We can still use that for the 4D space-time version with the understanding that the 0th (time-like) component of momentum is energy, and flow in the 0th direction means "density".

Thus, T_uv, is the flow of the uth component of 4-momentum in the vth direction.

T_00 is the flow of the 0th component of momentum, which is energy, in the 0th direction, which is density. Thus this is the energy density.

T_0i, (i = 1, 2, 3) is the flow of energy in the three spatial directions.

T_i0, is the density of each spatial component of momentum, and the lower

3x3 part retains the same meaning.

Now, it turns out that T_0i = T_i0. (The momentum density has to equal the energy flux, which can be seen from some simple mass flow considerations if you think about it). And likewise the lower part must be symmetric as well.

-Richard

Cougar

2010-Jan-23, 01:55 AM

...AND when you sum up all of that stuff ... what are we really talking about? We are talking about the effects of a different forms of energy at a very specific point in time and space?

Well, the point is, it works for any point.* :)

Isnt the net... really just energy in the end?

Apparently that's how nature 'feels' it. But (most?) all these different 'forms of energy' operate by different 'mechanisms', hence the necessary mixture of formulae.

____________________________

* With minimal exceptions.

tommac

2010-Jan-25, 12:38 AM

Cool ... thanks ...

There's a rather cute little way think of the components of the stress-energy tensor. The lower 3x3 part, the spatial (or space-space) parts are just the familiar 3D stress (momentum flux) tensor. That is the component T_ij is the flux of the ith component of momentum in the jth direction.

We can still use that for the 4D space-time version with the understanding that the 0th (time-like) component of momentum is energy, and flow in the 0th direction means "density".

Thus, T_uv, is the flow of the uth component of 4-momentum in the vth direction.

T_00 is the flow of the 0th component of momentum, which is energy, in the 0th direction, which is density. Thus this is the energy density.

T_0i, (i = 1, 2, 3) is the flow of energy in the three spatial directions.

T_i0, is the density of each spatial component of momentum, and the lower

3x3 part retains the same meaning.

Now, it turns out that T_0i = T_i0. (The momentum density has to equal the energy flux, which can be seen from some simple mass flow considerations if you think about it). And likewise the lower part must be symmetric as well.

-Richard

tommac

2010-Jan-25, 12:42 AM

Well, the point is, it works for any point.* :)

Yeah I understand that ... Also I can see why the calculations can get very complex when there are different influences from different energy sources.

Apparently that's how nature 'feels' it. But (most?) all these different 'forms of energy' operate by different 'mechanisms', hence the necessary mixture of formulae.

____________________________

* With minimal exceptions.

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