Tuesday, August 7, 2018

Musings On General Relativity Degrees of Freedom

The Einstein Field Equations in four dimensions consist of ten equations, each corresponding to elements of a symmetric tensor. One can show that given certain identities that this constitutes two degrees of freedom.

One can show that this is the same as the number of degrees of freedom for a spin-2 massless graviton is also two.

But, is this correspondence sensible?

The Einstein Field Equations, if you treat the stress-energy tensor as the source of the the tensor on the left hand side from which gravitational effects are derived, has the following sixteen components:

1. Mass (strictly speaking, energy density)
2. Electromagnetic flux in three spatial dimensions.
3. Linear momentum in three spatial dimensions.
4. Angular momentum in three spatial dimensions.
5. Pressure in three spatial dimensions.
6. Shear stress in three spatial dimensions.

This made all sorts of sense in the early 1900s when those were the only sources of mass and energy known.

But, a century later, this is weird and we know better. 

The Einstein Field Equations completely disregard both the strong force and the weak force that we know believe to be fundamental. But, we no longer think of pressure or shear stress as fundamental quantities, so what is it doing in a fundamental law of Nature?

For that matter, we usually think of electromagnetic flux in terms of photons at a fundamental level, not as two interrelated classical fields (a distinction with a difference without which, for example, transistors wouldn't work).

Also does it continue to make sense to formulate this with a hydrodynamic conception of the flow of matter and energy, when all of the other comparable laws of Nature are formulated in terms of point masses, which makes a certain amount of sense when you have a Standard Model made up of point masses.

Viewed in this light, why should a basically fundamental quantity, the degrees of freedom of a spin-2 massless graviton, have any correspondence to the Einstein Field Equations which omit entirely reference to some things that we think of as fundamental that gravitate, while elevating other quantities that we otherwise don't think of as fundamental to fundamental status?

And, if the components of the Einstein Field Equations seem rather arbitrary by modern standards, why should we assume that they really correspond to a spin-2 massless graviton?


neo said...

interesting read

what are you proposing instead

Mitchell said...

The electromagnetic field plays no essential role in the formulation of general relativity. The stress-energy tensor can be defined for any combination of fields.

andrew said...

@neo I'm not proposing an alternative other than to note a potential issue which may make GR as currently formulated less sacred.

@Mitchell While the stress energy tensor can be defined for any combination of fields, it isn't, and if it is defined for a different combination of fields, it isn't obvious to me that it would have the same number of degrees of freedom.

Mitchell said...

@andrew Please check out a textbook or serious introduction to general relativity. GR is not based on any specific assumption as to what the non-metric fields are. The equation "Einstein tensor proportional to stress-energy tensor" is still the basic equation of GR, whether or not the non-metric fields are absent, are just a Maxwell field, or are all the fields of the standard model. And in each case the stress-energy tensor involves gradients of energy-momentum density.

andrew said...

I will review the hefty GR textbook I have ("Gravity").

I don't disagree that the equation "Einstein tensor proportional to stress-energy tensor" is still the basic equation of GR, or that "in each case the stress-energy tensor involves gradients of energy-momentum density."

But, it is not obvious to me that the number of components in the stress-energy tensor does not influence the number of degrees of freedom in the field which is relevant to how you extrapolate from classical GR to potential QG theories.

Also, while I agree that assumptions about the non-metric fields are not axioms that were important to the development of GR, I have never seen the non-metric fields formulated in any other manner in GR. It is not obvious to me that the way that they are formulated is an irrelevant detail in the way that, for example, a choice of units for the physical constants involved, or a coordinate system, would be. It may be that they really aren't, but I can't easily see that this is true.