Geodesic Paths in Spacetime: How Mass and Velocity Affect the Curvature

[duordi]

Two masses A and B are accelerated to a velocity V on close “parallel” paths.

They will experience gravity or free-fall toward one another.

If I take the limit as mass A and B are reduced to zero and the velocity A and B is accelerated to approach the speed of light, ( I am approaching light photon conditions for A and B ), will the objects ( photons ) A and B attract one another / will their paths bend toward one another / will their paths cross one another

 

[RandallB]

You don’t need to take limits of mass somehow converting into photons. Photons and mass are different. You can just consider individual photons or particles of mass they will behave the same.
The problem you will have is when you define how the “bend” in your straight lines need to be it will likely be much smaller than your ability to define straight or parallel lines.
It would be like expecting a beam of collimated light to remain the same width forever, light doesn’t work that way.

 

[pervect]

Two masses A and B are accelerated to a velocity V on close “parallel” paths.

They will experience gravity or free-fall toward one another.

If I take the limit as mass A and B are reduced to zero and the velocity A and B is accelerated to approach the speed of light, ( I am approaching light photon conditions for A and B ), will the objects ( photons ) A and B attract one another / will their paths bend toward one another / will their paths cross one another

No, they will not. (I used to have some more specific references on this point, which can also be stated by saying that parallel light beams won't attract each other gravitationally. Note that anit-parallel light beams will attract each other).

 

[duordi]

Am I off base yet?

This makes sense to me when I consider that light would only have “relativistic mass” that only has an affect on an object in the direction of motion. So photons traveling in a perpendicular path would not gravitationally affect one another as they have no velocity perpendicular to the path of travel.

Am I correct in assuming that two photons on parallel paths passing one another from opposite directions would experience a gravitational interaction?
I would expect this because they do have a velocity component towards or away from each other at some point during the passing process.

How is this for a definition.

If P is a plane through photon A and perpendicular to the velocity vector of photon A then all photons traveling perpendicular to plane P and contain in plane P will not have a gravitational interaction with photon A.

 

[RandallB]

This makes sense to me

What does?
My comment that they will, but you will not be able to show it.
Or that it they will not.

You are working on individual particles (photons) not beams of particles.
(Also, I don't know what "anit-parallel" means, I assume it does not just mean perpendicular)

 

[duordi]

I was responding to Pervects' post.

Light which is traveling in Parallel and adjacent paths do not gravitationally attract because relativistic mass is directional. That is to say it requires a velocity with respect to an observer to exit. Light has no velocity perpendicular to the direction of travel and therefore has no relativistic mass in the direction perpendicular to travel.

 

[RandallB]

I was responding to Pervects' post.

Light which is traveling in Parallel and adjacent paths do not gravitationally attract because relativistic mass is directional. That is to say it requires a velocity with respect to an observer to exit. Light has no velocity perpendicular to the direction of travel and therefore has no relativistic mass in the direction perpendicular to travel.

Isn’t this contrary to standard GR assumptions that massless photon particles of energy should follow the same paths as traditional particles of mass would?
Thus, light bends around the sun just as a particle of mass would do.

I assume you expect two particles of mass starting off on your trajectory would not remain parallel as they curve together. Yet here the rule of photons behaving gravitationally the same as regular particles is not followed for some reason.
The reason for not retaining this GR assumption is not well explained IMO.

 

[duordi]

Two mass particles would follow a similar path as the photon described.
If two masses are traveling together they have no relative velocity with respect to one another and would therefore only observe the rest mass of the adjacent particle.

If the particles are small then the rest mass gravitational attraction will be small.

If however the same two masses pass each other at a velocity close to the speed of light then either of the particles will observe the other particle as having a large velocity and attribute relative mass with this velocity.

If the differential velocity approaches the speed of light and the particle mass quantities approach zero then the paths will approach a path identical to a path photons would follow.

A lot of odd things happen when the speed of light is approached.

 

[JesseM]

Isn’t this contrary to standard GR assumptions that massless photon particles of energy should follow the same paths as traditional particles of mass would?
Thus, light bends around the sun just as a particle of mass would do.

Are you sure this is a "standard GR assumption"? Particles of mass certainly don't follow the same geodesics through spacetime as photons, so I'm not so sure their paths through curved space would look the same either, although they would both obviously be deflected in some way by the Sun.

 

[RandallB]

Are you sure this is a "standard GR assumption"? Particles of mass certainly don't follow the same geodesics through spacetime as photons, so I'm not so sure their paths through curved space would look the same either, although they would both obviously be deflected in some way by the Sun.

Of the two potential concepts:

1) “Particles of mass certainly don't follow the same geodesics through spacetime as photons” - - should mean GR does not define the same GR curved space path for them.

2) “they would both obviously be deflected in some way by the Sun” - - They being photons vs, particles of mass; plus they would not be deflected just “in some way” but in exactly the same way, just as various values of small mass would all be deflected in exactly the same way by the Sun.

The only accepted and tested elements of GR assumptions I’m aware of confirm item #2.

And IMO if #2 is true it implies that #1 is false.

I’m not aware of any well founded and tested assumptions based on GR that indicate #1 could be true. If you have references to such I’ll look.

I suspect the reason is “space-time” was never and is not a good foundation from which the complete GR Theory was or could be built upon.
E.G. Einstein never gave “space-time” any real credit beyond being able to excite the interest of the general public that could not understand the details of something like GR. But he did feel that being able to involve the public at level they could appreciate was a good thing for science.

 

[JesseM]

1) “Particles of mass certainly don't follow the same geodesics through spacetime as photons” - - should mean GR does not define the same GR curved space path for them.

I don't understand what you mean by "GR does not define the same GR curved space path for them". Are you disputing the statement of mine that you quoted, or not?

I’m not aware of any well founded and tested assumptions based on GR that indicate #1 could be true. If you have references to such I’ll look.

You mean, references for the fact that different particles follow different geodesics depending on their velocity, and in particular that the geodesic path of light is different from the geodesic path of a massive particle? You might try looking up the difference between "timelike geodesics" and "null geodesics" (or 'lightlike geodesics') for starters. But are you doubting that two test particles can cross each other's paths at a single point in spacetime and yet follow different geodesics because their velocity at that point is different? If two particles are in different orbits around a planet but the orbits are elliptical so they cross paths at some point, do you doubt that both particles are both following geodesics even though the two paths go in different directions from the crossing-point?

I suspect the reason is “space-time” was never and is not a good foundation from which the complete GR Theory was or could be built upon.

What are you talking about? The whole point of GR is that it tells you the curvature of spacetime based on the distribution of mass and energy, and test particles follow geodesic paths in spacetime which maximize the proper time along that path (for timelike geodesics), just like how the inertial twin in the flat-spacetime twin paradox follows a geodesic path between two events on his worldline which causes him to age more (larger proper time) than his twin who follows a non-inertial path between the same two events.

E.G. Einstein never gave “space-time” any real credit beyond being able to excite the interest of the general public that could not understand the details of something like GR.

This is a very ignorant statement. Again, the basic tensor equations of GR (i.e. the Einstein field equations) are all about how matter and energy curve the 4D spacetime manifold, not how they curve space. For example, see this page from an introduction to Einstein's GR equation which says:

Similarly, in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. If you toss a ball, it follows a parabolic path. This is far from being a geodesic in space: space is curved by the Earth's gravitational field, but it is certainly not so curved as all that! The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where c=1. This allows a slight amount of spacetime curvature to have a noticeable effect.