Is the Accelerated Expansion of the Universe Misunderstood?

[Garth]

Space-time does not imply that the past the future and the present is all written out in one 4D space. In space-time we can speak just as much of a future, present and past.

Really? Where then is the 'present' in a space-time diagram?

Time warps just like space does.

How do you measure it - except as one observer observing the time dilation of a physical process in another frame of reference? Use of precise words is important to avoid confusion.

 

[Jorrie]

The FLRW metric has space curvature but no spacetime curvature.

Does that make any sense?
From my apparently flawed understanding of GR it does not at all.
Space is simply an observer dependent view on space-time is it not? Space in the absolute sense does not exist, at least according to GR, space cannot be treated separately from time. Or am I completely wrong about that?

Ok, maybe I should have stated: The FLRW metric has space curvature but no time curvature. And maybe you are wrong, maybe I am! But consider the following:

In local, inhomogeneous space, we choose a convenient reference frame - say one with a black hole permanently at rest at the origin. Then we treat the space around it as somewhat 'absolute', don’t we? At least, we have a fixed reference frame in such a case.

In cosmology, we choose a fixed reference frame in which the universe at large is static, at least in co-moving coordinates - better stated, we chose our frame so that the universe looks isotropic on the large scale. This is also somewhat of an 'absolute' space, which may be curved or not.

Further, everywhere in this homogeneous and isotropic space, we have the same cosmological time and no time differences from point to point. We may philosophize that this whole coordinate system's time is dilated relative to some or other 'cosmic time', where there is just empty space, no matter, energy and so on... But this is irrelevant – we do not have such a reference system.

Zooming in on our local space and time, we find space near massive bodies to have additional curvature (over and above possible large scale curvature) and we find time there to be dilated relative to the cosmological time of the large-scale metric. We call this local effect curved spacetime, where things fall under gravity.

It is not common to call the (homogeneous) large-scale effects curved spacetime. There things do not fall under gravity in the same sense. In fact, the metric ignores any spatial movement and things are just carried along with cosmic spatial expansion.

I’m aware of the dangers of expressing things only in words – words are open to interpretation. That’s why we prefer mathematical statements. But I’ll leave that to the PF mentors.

 

[marcus]

Hi Jorrie, ...
I find it puzzling that you say zero spacetime curvature and nonzero space curvature.
I would say it is typically the other way round: nonzero spacetime curvature and zero space curvature.

If you would care to, please give a reference link, or explain to me why I'm wrong

Hi Marcus.
I was referring to the FRW metric, where the constant coefficient of the temporal part indicates no time dilation, while the spatial part indicates space curvature, in general. I interpret this that the FRW metric can have space curvature but no spacetime curvature. It may be a case of wrong semantics though – maybe I should have said no time curvature, but that has a wrong ring to it!

I find your statement:"nonzero spacetime curvature and zero space curvature" equally puzzling!

Jorrie, thanks for your courteous reply---recognizing the possibility of simple differences in semantics.

I believe that a typical case of FRW metric has zero space curvature and NONZERO SPACETIME CURVATURE.

A way to see the nonzero spacetime curvature is to make a LOOP in spacetime and do parallel transport of a tangent vector. When you visualize this you will see that after going around the closed loop and returning to the starting point, the tangent vector will be pointing different.

this is essentially what nonzero spacetime curvature means----what spacetime NON-FLATNESS means.
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I will go over this in more detail:
As for the spatial curvature, it does not HAVE to be zero, that is just a very common case that people study. You have this spatial curvature parameter k which you can put equal -1, or 0, or +1. And it occurs in the Friedmann equation which governs how the FRW evolves.

It doesn't matter what we pick for k, because we are not concerned with the spatial curvature. So for SIMPLICITY let's consider the k=0 spatially flat case. Let us just take a vanilla case of FRW expanding universe!

Now imagine we have a time-machine-cum-spaceship, so that we can actually make a loop in spacetime------this is a mathematicians way of testing for curvature.
We are in galaxy A and we simply go BACK IN TIME along the worldline of galaxy A, for say a billion years. For simplicity imagine that the tangent vector we carry simply points along the worldline---it points in galaxy A time-direction. It will still do so when we have gone back in time a billion years (along the worldline geodesic). Then we take a little space-wards trip over to galaxy B. Say it is comparatively close but still far enough to be gravitationally loose from A, so it can drift apart. Because of comaparative closeness, the tangent vector we are carrying is hardly affected. Now we go FORWARDS IN TIME for a billion years along the worldline of galaxy B.

NOW we are in 2006 AD but in galaxy B, which has been carried far away from galaxy A by the expansion of space and we have this friggin tangent vector which we have to parallel transport back home to galaxy A. but it points nearly along the worldline of galaxy B and so, when we get it back home, it is going to be WAY SKEW. it will now have a SPATIAL VELOCITY COMPONENT that is "sideways" reflecting the recession speed of B relative to our home galaxy.

So when we complete the loop, the tangent vector we have been transporting around the loop is pointed all different from what we started with!

this is what curvature means. You can check it in 2D if you imagine yourself on the earth-ball at the equator and carry a vector originally pointing north around a loop-----say from equator up to north pole and then down another longitude line to the equator and then back along the equator. The tangent vector will no longer point north when you get back home and that is what it means for the surface of the Earth to have nonzero curvature.

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Jorrie you mentioned "time curvature" which doesn't mean anything to me. Spatial curvature is defined using 3D tangent vectors, tangent to a spatial section. Spactime curvature is defined using 4D tangent vectors which are tangent to the whole 4D manifold. There is no surrounding 5D space, so you study curvature INTRINSICALLY by groping around in loops like an ant crawling on a ball. Intrinsic is the only way and it is also in a certain sense elegant, because it doesn't need anything extra. you can FEEL the curvature by cruising around and feeling how your gyroscope twists and turns. that is what parallel transport is.
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So anyway, Jorrie, I think you better should not say that FRW has zero spacetime curvature (as in your post #19)
And probably not to talk about "time curvature". Spacetime curvature is a real and important thing. Good to know about. The fact that a typical spatially flat FRW metric is EXPANDING actually MEANS that it will have nonzero spacetime curvature. HTH.

 

[Jorrie]

Hi Markus,

Thanks for the very enlightening reply! It cleared up some important things for me, e.g., how the expansion creates spacetime curvature even in the absence of space curvature (k=0). There's just one small point of confusion lef!

Hi Jorrie, ...
I find it puzzling that you say zero spacetime curvature and nonzero space curvature.
I would say it is typically the other way round: nonzero spacetime curvature and zero space curvature.

After your last post, I understand and agree about spacetime curvature being non-zero, but why did you originally say space curvature is zero? The FRW metric specifically allows for k to be -, 0 or +. So k=0 is a very special case. I understand that it is common to take the k=0 for the model because our universe appears to be like that, but that unqualified statement was not right!

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So anyway, Jorrie, I think you better should not say that FRW has zero spacetime curvature (as in your post #19)

Agreed.

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…. And probably not to talk about "time curvature".

I agree and did indicate that in a previous post, i.e., ".. – maybe I should have said no time curvature, but that has a wrong ring to it!".
I fully agree.

Thanks again!

 

[Jorrie]

...
It is not common to call the (homogeneous) large-scale effects curved spacetime. There things do not fall under gravity in the same sense. In fact, the metric ignores any spatial movement and things are just carried along with cosmic spatial expansion.

I’m aware of the dangers of expressing things only in words – words are open to interpretation. That’s why we prefer mathematical statements. But I’ll leave that to the PF mentors.

Marcus has expertly proved my above statement about ‘no curved spacetime in the FRW metric’ to be wrong. See his post #33. Essentially, it means that as long as space is expanding or contracting, even with zero spatial curvature, spacetime is still curved.

But, as indicated in my post #32, I still think it cannot be looked upon as equivalent to local spacetime curvature effects, where curved spacetime goes hand-in-hand with gravitational time dilation and curved space.