Exploring Physical Connections in Einstein's Field Equations

[Herbascious J]

While viewing a recent lecture on Einstein’s Field Equations, the presenter made the association between the various quantities like mass, energy, momentum and pressure directly to the four dimensions of spacetime. Depending on which derivation of the FE he was explaining, he would make an association, for example “the time-time left side of the equation is associated with the mass quantity on the right side of the equation.” (I may have the wrong associations here). Other examples associated time-Xspace dimensions with something like momentum. I understand that there are ten equations that derive from the core FE, because out of the 16 possible, 6 of them are repetitive leaving 10. And I understand that there are many contributions to the stress-energy tensor side of the equation, but I have to admit I was a little surprised to see him associating different types of quantities with different combinations of physical dimensions. My question is this; is this association arbitrary, or are they somehow important on a physical level? Is there some laymen explanation that can show why these associations are being made? Is there some deep meaning behind these connections that wasn’t obvious before Einstein? I was always a little baffled as to why pressure was important in the FE because I always just interpreted it as a form of energy and why was it being high-lighted and singled out while something like chemical energy wasn’t, perhaps this is why. I realize this is tremendously complicated, but I was wondering if there was some physical insight that connected dimensions and physical quantities in these specific ways.

 

[PeterDonis]

While viewing a recent lecture on Einstein’s Field Equations

Can you give a specific reference?

I was wondering if there was some physical insight that connected dimensions and physical quantities in these specific ways.

Associations like this depend on a particular choice of coordinates. The simplest case is where we have chosen rectangular coordinates in a local inertial frame in which a particular observer is (at least momentarily) at rest (which is probably the choice the lecture you watched was implicitly using); for this case:

$T_{00}$ (the "time-time" component) represents the energy density measured by this observer;

$T_{0i}$ (the "time-space" components) represents the momentum density/energy flux measured by this observer;

$T_{ii}$ (the diagonal "space-space" components) represents the pressures in the 3 spatial directions measured by this observer;

$T_{ij}$ (the off-diagonal "space-space" components) represents the shear stresses measured by this observer.

 

[Herbascious J]

Here is the video I mentioned for reference.
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[Dale]

My question is this; is this association arbitrary, or are they somehow important on a physical level?

The association is not arbitrary. The stress energy tensor represents the flux of the four momentum. The four momentum consists of a time component (energy) and a space component (momentum). Energy can flow in the time direction which gives energy density as the time time component. Momentum can flow in the time direction or equivalently energy can flow in the space direction, this gives momentum density or equivalently energy flux as the time space component. Finally, momentum can flow in the space direction which gives stress as the space space component.

 

[Herbascious J]

The association is not arbitrary. The stress energy tensor represents the flux of the four momentum. The four momentum consists of a time component (energy) and a space component (momentum). Energy can flow in the time direction which gives energy density as the time time component. Momentum can flow in the time direction or equivalently energy can flow in the space direction, this gives momentum density or equivalently energy flux as the time space component. Finally, momentum can flow in the space direction which gives stress as the space space component.

If I am understanding correctly, each of these time-space or time-time (I think there are 10 combinations), all have an equivalent physical description that can be identified, for example mass, or momentum, pressure, etc. and these identifications are then desribed on the right hand side of the equation? I'm assuming they naturally fall out of the equations. And then we'd have ten unique equations? (I'm looking at PeterDonis' comment above.)

 

[Nugatory]

And then we'd have ten unique equations?

It’s more complicated that that because the tensor on the other side of the equation is the Einstein tensor, and that is seriously non-trivial: $\textbf{R}-\frac{1}{2}\textbf{g}R$ where $\textbf{R}$ is the Ricci tensor, $R$ is the Ricci scalar, and $\textbf{g}$ is the metric tensor. The Ricci tensor and constant are both calculated from the Riemann tensor, and Riemann's components are very complicated functions of derivatives of the components of the metric tensor. So when the dust settles, we have a set of non-linear coupled differential equations whose solution is the components of a metric tensor, but which are too hard to solve except for a few particularly simple stress-energy tensors.

At the same time, this complexity means that even simple stress-energy tensors can lead to interesting solutions. For example, black holes and gravitational waves are both solutions for a pure vacuum in which $T=0$; in this case we have $G_{\mu\nu}=8\pi{T}_{\mu\nu}=0$ but it does not follow that $R_{\mu\nu}$ and $R$ are zero.

 

[PeterDonis]

in this case we have $G_{\mu\nu}=8\pi{T}_{\mu\nu}=0$ but it does not follow that $R_{\mu\nu}$ and $R$ are zero.

Yes, it does; both the Ricci tensor and the Ricci tensor must be zero if the Einstein tensor is zero. (The simplest way I know of to see why is to take the trace of the EFE, which shows that $R = 0$, and then it's obvious from the form of the Einstein tensor that $R_{\mu \nu} = 0$.) What doesn't follow is that $R^\mu{}_{\nu \alpha \beta}$, the Riemann tensor, is zero.

 

[Nugatory]

Yes, it does; both the Ricci tensor and the Ricci tensor must be zero if the Einstein tensor is zero. (The simplest way I know of to see why is to take the trace of the EFE, which shows that $R = 0$, and then it's obvious from the form of the Einstein tensor that $R_{\mu \nu} = 0$.) What doesn't follow is that $R^\mu{}_{\nu \alpha \beta}$, the Riemann tensor, is zero.

Ah - right, yes it’s the non-zero Riemann that makes for an interesting spacetime when $T=0$

 

[pervect]

While viewing a recent lecture on Einstein’s Field Equations, the presenter made the association between the various quantities like mass, energy, momentum and pressure directly to the four dimensions of spacetime. Depending on which derivation of the FE he was explaining, he would make an association, for example “the time-time left side of the equation is associated with the mass quantity on the right side of the equation.” (I may have the wrong associations here). Other examples associated time-Xspace dimensions with something like momentum. I understand that there are ten equations that derive from the core FE, because out of the 16 possible, 6 of them are repetitive leaving 10. And I understand that there are many contributions to the stress-energy tensor side of the equation, but I have to admit I was a little surprised to see him associating different types of quantities with different combinations of physical dimensions. My question is this; is this association arbitrary, or are they somehow important on a physical level? Is there some laymen explanation that can show why these associations are being made? Is there some deep meaning behind these connections that wasn’t obvious before Einstein? I was always a little baffled as to why pressure was important in the FE because I always just interpreted it as a form of energy and why was it being high-lighted and singled out while something like chemical energy wasn’t, perhaps this is why. I realize this is tremendously complicated, but I was wondering if there was some physical insight that connected dimensions and physical quantities in these specific ways.

Pressure * volume has units of energy, but it's not the same as energy. In particular, if you imagine an incompressible fluid, applying pressure to the fluid element, as judged by it's rest frame the process of pressurizing the fluid doesn't store any energy in it.

What makes the issue tricky to understand is the relativity of simultaneity, which first arises special relativity.

Very similar issues arise in special relativity when one tries to calculate "the energy" or "the mass" of a section of a rod under pressure. Rindler performs a relevant calculation, calculating the ratio of net force / acceleration of a section of pressurized rigid rod, in his book "Relativity, special, general, and cosmological". His technique is a bit old fashioned, but the results are correct. The ratio of force/acceleration, which Rindler calls "mass", depends on the pressure in the rod. Understanding the "why" of this takes some time and study, but it may help you understand the "why" in the more complex situation of GR, where pressure can also affect the "mass" of a system, suitably defined.

A GR extension of Rindler's simpler calculation would be to illustrte that the ratio of force/acceleration of a Born-rigid rod being towed by one end, the force measured by a local scale, depends on the stress distribution in the rod. However, I've never seen this problem worked out in detail.

One can ultimately trace back the issue to the hidden influence of the relativity of simultaneity, but I believe an extended discussion of this may be premature.