The Nature of Time

[Mordred]

4 hours ago, Genady said:

Agreed. This detour was just a little exchange between @sethoflagos and me and should've been over with the @Markus Hanke's post above. However, some posters have picked up some comments as if they were standalone statements and have used them to educate the audience in an unrelated material.

Unfortunately it does relate in one regard as a common misconception is that a higher density past equates to spacetime curvature and hence time dilation with regards to the FLRW metric. However this isn't the case there is subtle differences in curvature with regards to the FLRW metric to spacetime curvature in GR with regards to time and proper time. Anyways that's likely best left off for a different discussion.

Edited January 12, 2023 by Mordred

[MigL]

Time is included in many models, and has subtle, but nonetheless different aspects.
It would be narrow minded to restrict a discussion on the nature of time, to a single, or few aspects, while ignoring the rest.

My opnion.

[Genady]

1 hour ago, sethoflagos said:

If we created an image of a star from not just its emr transmissions but all concurrent arrivals of its cosmic rays in a range of velocities, we'd obtain a trace of it's historic worldline as viewed from our current location in space.

That is what we see on a starry sky. We see one picture that includes stars as they were just several years ago and stars as they were thousands of years ago, all at once. Another example is multiple images of a galaxy created by its light going around a massive object by different routes. As these routes are of different lengths, we see at once the galaxy as it was at different times in the past.

[Mordred]

41 minutes ago, MigL said:

Time is included in many models, and has subtle, but nonetheless different aspects.
It would be narrow minded to restrict a discussion on the nature of time, to a single, or few aspects, while ignoring the rest.

My opnion.

True enough considering all the different variations coordinate time, proper time, conformal time, commoving time. Though in each case it's more accurate to treat these as defined observers.

[Markus Hanke]

20 hours ago, studiot said:

I would like to see the mathematics supporting this statement.

Misner/Thorne/Wheeler “Gravitation” describes this in some detail. Consider first the general form of the FLRW metric:

\[ds^{2} =-dt^{2} +a( t)^{2} d\Sigma ^{2}\]

wherein \(\Sigma\) designates a 3-surface of uniform curvature (which could thus be elliptical, Euclidean or hyperbolic). The full Riemann tensor for 3D+1 spacetime with this type of metric then has six functionally independent components:

\[R_{1100} =R_{2200} =R_{3300} =a\ddot{a}\]

\[R_{1122} =R_{2233} =R_{3311} =-a^{2}\dot{a}^{2}\]

Thus spacetime is never Riemann-flat, unless a(t)=const. 

On the other hand, the 3D+0 Riemann tensor for a given 3-surface \(d\Sigma\) of space is

\[^{3}R_{ijkl} =\frac{k}{a( t)^{2}}( g_{ik} g_{il} -g_{il} g_{jk})\]

wherein k is called the curvature parameter, so that for the choice k=0 each 3-surface is Euclidean and flat. MTW motivates the presence of the curvature parameter in this expression by using the following exact solution of the field equations:

\[d\tau ^{2} =-dt^{2} +a( t)^{2}\left(\frac{du^{2}}{1-ku^{2}} +u^{2} d\phi ^{2}\right) ;\ u=\frac{r}{a( t)}\]

wherein t is the total time recorded on a co-moving “dust clock” since the beginning of the universe.

[studiot]

On 1/13/2023 at 10:28 AM, Markus Hanke said:

Misner/Thorne/Wheeler “Gravitation” describes this in some detail. Consider first the general form of the FLRW metric:

etc

 

Thank you so much for making this effort on my behalf. +1

 

I have been looking for a resonably priced copy of MTW for some time, bu noone seems to want to pass theirs on.

As luck would have it, I found one (hardcopy no less) of a 2021 version new but shop soiled at a bearable price.

Fantastic but a lot to wade through as I do not know the book.

In particular I could not find reference to the FLWR metric in the index so I wonder if you could oblig by beefing up your reference. !300 pages is alot to read through to find something.

[Markus Hanke]

8 hours ago, studiot said:

In particular I could not find reference to the FLWR metric in the index so I wonder if you could oblig by beefing up your reference. !300 pages is alot to read through to find something.

Check part VI, chapter 27.5-6. In my copy that is page 718 onwards, where they talk about the expansion factor and the possible geometries of hypersurfaces of homogeneity. The expressions for the non-zero components of Riemann for FLRW come from my notes, which, if memory serves right, I got from MAPLE at some point.

8 hours ago, studiot said:

Fantastic but a lot to wade through as I do not know the book.

It’s considered the gold standard so far as the theoretical aspects of GR are concerned for a reason. Truly fantastic text, most of my own GR knowledge comes from here. There are even some bits which are not found in any other GR text that I’m aware of.