Breakdown of Correspondence Principle: Null Dust Case

[victorvmotti]

In both quantum and general relativity theories we are used to provide results in the "limited" conditions to demonstrate a correspondence between new and old formalism.

For instance deflection of light of a star due to Sun in GR is double the amount given in classical theory.

Yet I have problems to sense a limit condition on the gravity due to pure radiation also called null dust solution of Einstein's field equation for a perfect pressureless fluid.

If we tend to zero both the masses on the Newtown gravity law the force among the objects/particles will be zero.

But based on Einstein's field equation the energy momentum tensor does not vanish in the case of massless objects and is defined in terms of two null four velocities which is the equation for a dust with zero pressure. Therefore the nonvanishing curvature tensor says that a group of massless particles have gravity among themselves.

Is this a correct understanding?

And is it a counter example to correspondence principal in GR?

 

[martinbn]

It is not enough to have small masses for the limit to norelativistic gravity. You also need the objects to have small velocities (and the stresses need to be smaller than the energy dencity).

 

[victorvmotti]

Yes you are right, my point is that a correspondence cannot be made given the nature of Newton law.

 

[PeterDonis]

If we tend to zero both the masses on the Newtown gravity law the force among the objects/particles will be zero.

In Newtonian gravity, yes. But you can't use Newtonian gravity to describe null dust, or in fact to describe anything "massless". That concept simply doesn't exist in Newtonian gravity.

the nonvanishing curvature tensor says that a group of massless particles have gravity among themselves.

Yes.

is it a counter example to correspondence principal in GR?

What "correspondence principle" are you talking about? Can you give a reference?

 

[victorvmotti]

I am trying to imagine what the cosmos looked like or its dynamics in the radiation dominated era after Big Bang. Thought that a correspondence with Newton law in the limits might be helpful.

Sean Carroll in his Spacetime and Geometry book adopts a sort of correspondence approach in Chapter 4, pp 151-159 where he argues if the new theory by Einstein predicts the Newtonian theory.

 

[bcrowell]

What "correspondence principle" are you talking about? Can you give a reference?

People more frequently use "correspondence principle" to refer to classical physics as compared to quantum mechanics, but the idea is broader. As applied to relativity, one way of expressing it is to say that relativity has to be consistent with Newtonian mechanics in the limit of weak gravitational fields and velocities much less than c.

For the OP, the problem is that, as pointed out by martinbn, the relevant limit does not apply here. A null dust is made of particles traveling at c, and that's the opposite of the Newtonian limit.