B Space-Time Expansion: Objects Accelerating Without Force?

[Martin Philpott]

I am familiar with the concept of the curvature of space-time. I imagine that space-time can also expand. If this is possible then objects far from our view may appear to accelerate. No force required they are traveling through space that is changing in curvature or expanding. Is this a possibility. I have not done the maths here, just a guess.

 

[ergospherical]

Spacetime cannot "expand" or change at all for that matter; the spacetime manifold is the entire history of the universe.

 

[Martin Philpott]

Oh I was thinking that the curvature of space-time actually explains the apperent 'gravitational attraction' that we observe. I think that I need a 2nd opinion.

 

[PeroK]

Oh I was thinking that the curvature of space-time actually explains the apperent 'gravitational attraction' that we observe. I think that I need a 2nd opinion.

The current cosmological model for the universe has space expanding as a function of time.

But, @ergospherical is correct, spacetime itself has no concept of evolution as time is an integral part of spacetime.

Yours is a common misconception that there is a second timelike parameter according to which spacetime evolves - or, at least, this is what you are imagining without realising it.

 

[PeroK]

I am familiar with the concept of the curvature of space-time. I imagine that space-time can also expand. If this is possible then objects far from our view may appear to accelerate. No force required they are traveling through space that is changing in curvature or expanding. Is this a possibility. I have not done the maths here, just a guess.

There's no need to guess or do the maths for yourself. You can read about the expanding universe, which has been the dominant cosmological model since the 1920s.

 

[Dale]

Oh I was thinking that the curvature of space-time actually explains the apperent 'gravitational attraction' that we observe. I think that I need a 2nd opinion.

He is making a distinction between spacetime and space. Spacetime includes time, so spacetime doesn't expand. Space expands as time evolves.

Think of a trumpet. It is narrow on one end and it is wide on the other end.

The circumference of the trumpet is "space" and the length of the trumpet is "time". So the trumpet is a shape where "space" expands over "time". But the trumpet itself ("spacetime") is not expanding.

 

[ergospherical]

yes and what is meant by the "expansion of space" is that the expansion $\theta = \nabla_a u^a$ of a congruence of geodesics of a cloud of non-interacting particles is not equal to zero (i.e. you would see the particles moving closer to, or further from, each other)

 

[Dale]

yes and what is meant by the "expansion of space" is that the expansion $\theta = \nabla_a u^a$ of a congruence of geodesics of a cloud of non-interacting particles is not equal to zero (i.e. you would see the particles moving closer to, or further from, each other)

Yes. In the trumpet analogy, if you draw several lines, each going straight down the length of the trumpet, you will see that the lines are closely spaced on one end and spaced far apart on the other end. This is the trumpet's "space" expanding over "time".

 

[pervect]

The physical significance of "expanding space" is often over-sold. Consider Minkowskii space, which can be regarded as the empty, flat, space-time of special relativity. However, a model mathematically equivalent to Minkowskii space exists, the so-called Milne model. See for example https://en.wikipedia.org/w/index.php?title=Milne_model&oldid=991667276. This Milne model has a "space" that "expands" with time. It is in fact a limiting case of the standard cosmological model with no matter, energy, or pressure.

Basically, the geometrical structure of Minkowskii space and the Milne model represents the same space-time geometry. Only the method of assigning time (and space) coordinates distinguish the two.

This is all a bit abstract, so let me give a hopefully more familiar example. A (flat) two dimensional plane has the geometry of a plane, this geometry is independent of the coordinates used. It doesn't matter to the geometry itself whether one uses polar coordinates or cartesian coordinates. The coordinates can be (and should be) regarded as labels, whose purpose is to identify points on the geometry.

There is a mathematical relationship, a mapping, in the plane that allows one to transform polar coordinates into cartesian coordinates, and vice-versa. Similarly, there is a mathematical transformation that maps the space-time coordinates of the Milne model to the (t,x,y,z) cartesian space-coordinates that one usually uses in special relativity. The existence of this mapping (also known as a diffomorphism), demonstrates that the two underlying geometries are identical. Superfically, the coordinates descriptions of the "expanding" Milne model and of the "non-expanding" flat space-time of special relativity appear different, but fundamentally they represent the same geometry. This example illustrates that "expansion" isn't a geometrical property, but a statement that depends on one's choice of coordinate labels.

 

[PeterDonis]

This example illustrates that "expansion" isn't a geometrical property, but a statement that depends on one's choice of coordinate labels.

More precisely, the sense of "expansion" you are using is coordinate-dependent. But there is a different sense of "expansion" which is not: the expansion scalar of a congruence of worldlines. The expansion scalar of the congruence of "comoving" worldlines in the Milne model has a positive expansion, regardless of any choice of coordinates. But of course in flat Minkowski spacetime there is another congruence of worldlines (the ones at rest in some standard inertial frame) that has zero expansion. The switch of coordinates you describe corresponds to switching the congruence of worldlines that we consider to be "at rest" in our chosen coordinates. But each congruence itself still has properties that are coordinate-independent, such as its expansion scalar.

 

[Vanadium 50]

Uh guys? B-level, remember?

 

[PeterDonis]

B-level, remember?

Yes, the term "expansion scalar of a congruence of worldlines" is not really B-level, so let me rephrase in B-level terms: if we pick a family of observers in a spacetime, we can ask whether they are moving away from each other or not. The Milne model is based on picking such a family of freely falling observers in flat spacetime that is moving away from each other; hence it is called "expanding". But one can also pick a family of freely falling observers in flat spacetime that is not moving away from each other, and the latter family is the most "natural" family of observers in flat spacetime, since they will all be at rest in the same inertial frame.