Coordinate time vs Proper time

[Alan Grayson]

Suppose the universe is a hyper-sphere, not expanding, and an observer travels on a closed loop and returns to his spatial starting point. His elapsed or proper time will be finite, but what is his coordinate time at the end of the journey?  TIA, AG

[Alan Grayson]

On Monday, May 18, 2020 at 3:28:40 PM UTC-6, Alan Grayson wrote:

Suppose the universe is a hyper-sphere, not expanding, and an observer travels on a closed loop and returns to his spatial starting point. His elapsed or proper time will be finite, but what is his coordinate time at the end of the journey?  TIA, AG

It's not a dumb question IMO. If you circumnavigate a spherical non-expanding universe, what happens to coordinate time at the end of the journey? Does something update the time coordinate? Or does it somehow miraculously(?) remain fixed? TIA, AG 

[Brent Meeker]

Are you supposing the universe is a 3-sphere?  In that case It's just like going around a circle.  The degree marks on the circle are coordinates, they have no physical meaning except to label points.  So if you walk around the circle you measure a certain distance (proper time) but come back to the same point.

Or are you supposing it's a 4-sphere so that all geodesics are closed time-like curves?  I don't know how that would work.  I don't think there's any solution of that form to Einstein's equations.

Brent

[Alan Grayson]

I'm supposing a 4-sphere and (I think) closed time-like curves. The traveler returns presumably to his starting position, but is the time coordinate unchanged? AG 

[Brent Meeker]

I don't think there's any very sensible answer in that case.  Goedel showed there can be solutions with closed time-like curves if the universe is rotating.  But solutions of GR don't have any dynamic connection to matter and the entropy of matter.  In the same spirit there could be a solution to quantum field theory that was close around the time like curve...in which case you'd experience "Groundhog Day"...including your thoughts.

Brent

[Alan Grayson]

What does entropy have to do with this problem? AG 

[Brent Meeker]

Increasing entropy points the direction of time.

Brent

[Alan Grayson]

Let me pose the question another way: Is coordinate time ever updated? AG 

[Alan Grayson]

Or say, in the Twin Paradox, the elapsed or proper time for the traveling twin is less than for the Earth-bound twin, but when they meet, do they share the same coordinate time? AG 

[smitra]

On 23-05-2020 23:01, 'Brent Meeker' via Everything List wrote:
> On 5/23/2020 11:51 AM, smitra wrote:
>
>>> My point is that identity is an intrinsic property of what
>>> something
>>> is now. The history of the of the constituent particles have no
>>> affect
>>> on the behaviors or operation of those particles. To say the
>>> history
>>> is relevant to identity is to add an arbitrary extrinsic property
>>> which can be of no physical relevance.
>>>
>>> This is a direct consequence of QM, you can't distinguish two
>>> electrons, from each other.
>>> But they still have locations and histories, c.f. Griffiths
>>> consistent
>>> histories interpretation of QM or Feynmann's path integral QM.
>>> When
>>> electrons make spots on the film in an EPR experiment the electron
>>>
>>> that made this spot is not identical with the electron that made
>>> that
>>> spot in the sense of being the same electron. And in any case I
>>> don't
>>> see how the sameness of particles implies the sameness of complex
>>>
>>> structures made of particles, i.e. persons.
>>>
>>> Brent
>>
>> Physics is local, all the relevant information to describe what I
>> feel right now is contained in my brain at this exact moment. While
>> this can all be explained in terms of information in the past, that
>> doesn't take away from the fact that it is also present right here
>> in my head. Also, not all the information was present in the past
>> state due to effective collapse of the wavefunction. In general, I
>> end up in a superposition of states which has the exact same
>> information content as the past state. I then find myself in one of
>> the possible components of such a superposition (in the
>> computational basis states of the classical algorithm that my brain
>> is running).
>>
>> Saibal
>
> In the MWI, there is never any increase in the total information. All
> evolution is unitary and reversible. Local information appears
> because event horizons make correlations (negative information)
> inaccessible.
>
> Brent

That's the case for the global wavefunction. But if you are in a
superposition of making a binary choice one way or the other with equal
amplitudes, then you will find yourself in one or the other part of the
superposition. So one bit of information appears in the sector you are
in which is essentially the information needed to specify in what sector
you are.

Saibal

[Brent Meeker]

In general relativity points are labelled by coordinates which locally smooth functions that define a Reimannian space.  You can change them to whatever you want whenever you want.  They are just labels.  I'm not sure what "update" means in this context.  Physical time is measured by clocks, and they measure proper time along their world lines.

Brent

[Brent Meeker]

Yes.  Coordinates are labels for points, so if you're together with your twin, you both are at the same point in spacetime and that point only has one label in any given coordinate system.

Brent

[Alan Grayson]

Since time is just of the 4 labels for spacetime points, can they be assigned at random? What specific function do they satisfy? AG 

[Alan Grayson]

Since time is just ONE of the 4 labels for spacetime points, can they be assigned at random? What specific function do they satisfy? AG 

How is the time coordinate chosen such that the Lorentz distance between spacetime points is meaningful? AG 

[Brent Meeker]

They're not assigned at random because they need to form a locally smooth flat tangent space so that Einstein's equations will apply.  And that's their function...they label in the points so that the equations apply and when you solve the equations you will know what value goes with what point.

Brent

[Brent Meeker]

The proper distance/duration is an invariant, it doesn't depend on the coordinate system.

Brent

[Alan Grayson]

I think the invariance of proper distance/duration a direct result of the Lorentz transformation, and is one of the results of SR. If that's the case, is it used in GR to derive EFE's? TIA, AG

[smitra]

The Lorentz transform results from demanding that ds^2 for a flat
space-time is an invariant. It's easy to derive this, as you know
rotations and translations leave the ordinary Euclidic metric invariant,
the relative minus sign between time and space means that instead of
cos(theta) and sin(theta), you get cosh(theta) and sinh(theta) in
transforms that mix time and space.

Saibal

[Alan Grayson]

But why would you want ds^2 to be invariant? The answer IMO, is that

the LT leaves the SoL invariant as we change coordinate systems, and

using this requirement is sufficient for deriving the LT. AG