Pressure and refraction in GR?

[cefarix]

I want to understand why pressure creates gravity under GR (not how, why). What kind of pressure are we talking about here? Is this just the potential energy of pressure acting as mass?

And the second part: I've been trying to model gravity as refraction of masses moving with constant velocity through a spacetime that has a varying density. Spacetime is more dense around masses, and other masses traveling through spacetime have their paths refracted due to the density changes, and so appear to fall "in" towards the mass causing the increase in density. Does this model of gravity concur with GR?

 

[Garth]

Welcome to these Forums cefarix!

First part. Pressure is a form of energy. Consider ordinary gas pressure that is caused by the motion of the gas molecules. As pressure increases with temperature so does the velocities and hence kinetic energy of the molecules. According to SR their relativistic mass or total energy increases with velocity, and according to GR that increased mass/energy increases the gravitational field of the medium under consideration.

Second part. Whatever do you mean by: "a spacetime that has a varying density"? What is the density of space-time??
Without a precise definition and detailed calculations it is impossible to say whether your model concurs or not with GR - but I doubt it.

Garth

 

[Crosson]

Notice that pressure has the same units as energy density.

In classical electromagnetism, the energy density of the EM field is identical to the pressure it is exerting. Remember that all pressures are due to electromagnetic forces.

In GR, energy density creates gravitation. So pressure (electromagnetic energy density) creates gravitation.

Does this model of gravity concur with GR?

It depends. If the "density of spacetime" is a second rank tensor which is a solution to Einstein's Field Equations, then maybe.

 

[cefarix]

I'm treating spacetime as a compressible medium, so by density I mean how compressed or rarefied spacetime geometry is, relative to the observer of course. The units of spacetime density then come out to be a unitless quantity actually, but its actually a ratio between the remote location's geometry and the observer's geometry (which is inherently 1:1 scaled).

Density of spacetime translates into geometry because spacetime is our "measuring stick". So, according to my model, near a black hole for example, the density of spacetime is changing very quickly as you get nearer to the black hole, i.e., the gradient is very steep. And so, a mass (or a light beam for that matter) moving near the black hole will have its path bent highly due to refraction.

I hope that makes you guys understand better what I mean by spacetime density?

And, um, I haven't studied tensor maths or GR formally, so I don't know much of the GR-related maths.

 

[pervect]

If you take a look at fluid dynamics, you might start with a number rho that describes the density of the fluid, a simple scalar function of position. This appears to be where you are trying to start from.

But a simple description of density vs position is not enough to describe the general state of a fluid. One must add in the fact that the fluid elements can be moving in any given direction - this is often done via the concept of streamlines.

So now one has a number rho at any point, and three numbers to describe the streamline at that point (the velocity components of the streamline in the x,y, and z directions will do).

One quickly find that even this is not a sufficient number of properties to classify the state of a fluid, for a fluid also exerts pressure, and this is yet another variable besides density and velocity. Pressure can exist in both moving and non-moving fluids.

When one wrap all of these effects up into one big convenient package, one comes up with the description of a fluid by a stress-energy tensor. The classical stress-energy tensor is a 3x3 matrix.

GR simply builds on these classical foundations, and describes the density and motion of matter via a stress-energy tensor. The stress-energy tensor is generalized from three dimensions to four, as GR is a theory of space-time, not just a theory of space. This extension is really fairly trivial, though.

The net result is a tensor which involves ten quantites that are defined at any point. The density of the fluid is one, the density multipled by the streamline velocity give three more (this is just the average momentum of the fluid at that point in space-time), and the pressures in the x,y, and z directions give three more. This is a somewhat simplified description of 7 out of the 10 quantities that exist at every point in space-time which are included in the stress-energy tensor. I'm not sure how to simply describe the other three at this point.

GR relates the stress-energy tensor to the curvature of space-time at that point, via the simple formula

G_uv = 8 Pi T_uv

The quantity on the left is the Einstein curvature tensor, 8 Pi is a constant, and T_uv is the stress-energy tensor of the matter / energy distribution.

It is unlikely that your formula will be the same as GR if you cannot show that it can be written in equivalent tensor notation. Probably you have not given much thought to how your solution transforms under a change of coordinate, for instance.




.

 

[Mortimer]

I've been trying to model gravity as refraction of masses moving with constant velocity through a spacetime that has a varying density. ... Does this model of gravity concur with GR?

This has been investigated by others. See e.g. http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:gr-qc/0107083
Jose Almeida is an optics prof at the university of Minho, Portugal. Not surprisingly his work is called 4D optics.