Are Geodesics in Outer Space Curved or Straight Paths?

[Mentz114]

In the case of the distant galaxy, we can't confirm that motion visually and according to GR we can't assign a relative velocity to it either, so how exactly do we attribute the redshift we observe to a radial motion if we can't independently confirm that motion?

According to Stephani ( "General Relativity" ( 1986)

"[formula] ... shows that the two effects can only be separated in an artificial manner depending on the coordinate system"

The 'two effects' being doppler shift and cosmological shift.

 

[TrickyDicky]

According to Stephani ( "General Relativity" ( 1986)

"[formula] ... shows that the two effects can only be separated in an artificial manner depending on the coordinate system"

The 'two effects' being doppler shift and cosmological shift.

I completely agree with this statement.

I'm not sure how it addresses my concern, I just don't know how to link the redshift we perceive (nevermind if you want to call it Doppler or Cosmological) with remote motion of galaxies, unless we proceed by elimination of alternatives that can't explain the redshift-distance relation.

 

[atyy]

I completely agree with this statement.

I'm not sure how it addresses my concern, I just don't know how to link the redshift we perceive (nevermind if you want to call it Doppler or Cosmological) with remote motion of galaxies, unless we proceed by elimination of alternatives that can't explain the redshift-distance relation.

The framework of cosmology is general relativity. A universe must be a solution of the Einstein field equations.

The red shift, plus other data, plus an assumption that the universe is almost uniform on large scales, constrain the universe to be an FRW-like solution of the Einstein field equations. The FRW solution is the "objective" thing corresponding to the "objective" red shifts that certain observers see. The description of the FRW solution as "expanding universe" is observer dependent, as there are many other descriptions of the FRW solution.

 

[TrickyDicky]

The framework of cosmology is general relativity. A universe must be a solution of the Einstein field equations.

The red shift, plus other data, plus an assumption that the universe is almost uniform on large scales, constrain the universe to be an FRW-like solution of the Einstein field equations. The FRW solution is the "objective" thing corresponding to the "objective" red shifts that certain observers see. The description of the FRW solution as "expanding universe" is observer dependent, as there are many other descriptions of the FRW solution.

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

 

[atyy]

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

Yes. Apart from experimental data like the red shift, the two major things used to eliminate are the Einstein Field Equations and the assumption of uniformity. The assumption of uniformity is called the "cosmological principle". It is possible that further observations may show this is not justified, since we apply it to the whole universe, even those parts we cannot observe (even in principle, if the Einstein equations are right, not just due to technological limitations). Some of those currently unobservable parts will become observable in the future.

http://arxiv.org/abs/1005.2165 (Solid speculation, it shows what's being worked on at the moment)

 

[Passionflower]

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

That seems the common view but for instance Penrose’s Weyl tensor hypothesis is interesting,e.g. FLRW is basically no Weyl curvature, but in time matter tends to clump due to gravitation, thus increasing Weyl, thus the model diverts from reality in time.

 

[JesseM]

If I understand this correctly after giving it some thought, you say that expansion or "relative motion" are clearly coordinate-dependent but "visual appearances", namely redshift, is coordinate-independent since it's a purely local physical fact, this leads to the question: how can we interpret a local coordinate-independent measure (redshift) as a sign of a coordinate-dependent motion (relative velocity of a distant galaxy) that is not in our local frame. I thought in GR we couldn't even define a relative velocity in a different inertial frame(due to path-dependent parallel transport, etc).

You can't define a coordinate-independent notion of relative velocity, but you can certainly talk about relative velocity in some choice of cosmological coordinate system. In fact if I understand things right you really only need a choice of simultaneity convention, then you can determine the "proper distance" to any object on a surface of simultaneity (which is just found by integrating the metric line element along a particular spacelike curve between the two objects that is entirely contained within that surface of simultaneity, just like you do to find proper time along a timelike worldline), and you can then talk about how the proper distance to some object is changing with your own proper time. The velocity given by Hubble's law can be understood in terms of change in proper distance over change in cosmological coordinate time, which is the same as the proper time of any galaxy at rest relative to the Hubble flow.

 

[TrickyDicky]

That seems the common view but for instance Penrose’s Weyl tensor hypothesis is interesting,e.g. FLRW is basically no Weyl curvature, but in time matter tends to clump due to gravitation, thus increasing Weyl, thus the model diverts from reality in time.

I also find the Weyl curvature hypothesis interesting, it seems to me it is trying to tell us something about the FRW model, not exactly sure if it is that it is diverting from reality, though.

 

[TrickyDicky]

You can't define a coordinate-independent notion of relative velocity, but you can certainly talk about relative velocity in some choice of cosmological coordinate system. In fact if I understand things right you really only need a choice of simultaneity convention, then you can determine the "proper distance" to any object on a surface of simultaneity (which is just found by integrating the metric line element along a particular spacelike curve between the two objects that is entirely contained within that surface of simultaneity, just like you do to find proper time along a timelike worldline), and you can then talk about how the proper distance to some object is changing with your own proper time. The velocity given by Hubble's law can be understood in terms of change in proper distance over change in cosmological coordinate time, which is the same as the proper time of any galaxy at rest relative to the Hubble flow.

Sure, that is what the FRW model does, give you that choice of coordinates and simultaneity convention and as we were saying you only need two things to reach that model, GR field equations and the assumption of spatial uniformity. Both are solid, the second one was at first a philosophical assumption, but within not many years it is close to be observationally settled.

 

[JesseM]

Sure, that is what the FRW model does, give you that choice of coordinates and simultaneity convention and as we were saying you only need two things to reach that model, GR field equations and the assumption of spatial uniformity. Both are solid, the second one was at first a philosophical assumption, but within not many years it is close to be observationally settled.

Yeah, but the choice of simultaneity used to describe the FRW model is still a matter of convention, you could in theory describe the metric of an FRW universe with a coordinate system that used a different choice of simultaneity (though it would be a lot less convenient), and the fundamental laws of physics don't pick out any global simultaneity convention as being "preferred" over any other.