Does cosmological time define an observer-independent global instant?

[OmniThoughts]

TL;DR

In "spacetimes resulting from the Big-Bang" where τ is regular, doesn't the cosmological time function supply how a global instant is not impossible? Regarding "the geometric slice: a τ-slice", is "the sharing of a τ value" a global, observer-independent shared relation that is fixed by the universe's spacetime geometry alone?

I've often seen it said that, due to Relativity of Simultaneity, relativity makes a global instant impossible. τ at an event is the supremum of proper time along past-directed causal curves from it (Andersson, Galloway and Howard 1998, Class. Quantum Grav. 15, 309). A global instant is a surface that "every worldline reaching that particular value of τ" crosses once, whose events aren't even causally related, and is defined without reference to any observer. In "spacetimes resulting from the Big-Bang" where τ is regular, doesn't the cosmological time function supply how a global instant is not impossible? Regarding "the geometric slice: a τ-slice", is "the sharing of a τ value" a global, observer-independent shared relation that is fixed by the universe's spacetime geometry alone?

 

[Dale]

I've often seen it said that, due to Relativity of Simultaneity, relativity makes a global instant impossible.

Can you provide a citation and a direct quote for this saying? The Andersson reference doesn't say this.

A global instant is a surface that "every worldline reaching that particular value of τ" crosses once, whose events aren't even causally related, and is defined without reference to any observer.

Do you have a source for this definition? "Global instant" is an unusual term that I have not heard of before. The Andersson reference doesn't define "global instant".

I don't see any reason that "global instants" (defined as you do) should be impossible in general relativity. You may be mixing up things from special relativity and general relativity. It is hard to tell with inaccurate citations.

 

[PeterDonis]

doesn't the cosmological time function supply how a global instant is not impossible?

Not in the sense you appear to mean.

In FLRW spacetimes, the spacetime does have a particular symmetry property, that it can be seen as an infinite stack of "slices"--spacelike 3-surfaces that can be viewed as "the universe at an instant of time", with each slice labeled by a unique value of "cosmological time". Each slice is homogeneous and isotropic, and there is a set of observers, called "comoving observers", whose proper time is the same as "cosmological time" (at least in the most common choice of time coordinate for "cosmological time"--but note that that's not the only possible one), and who always see the universe as homogeneous and isotropiic.

However, none of this constitutes a "global simultaneity" or defines a "global instant" in the sense you appear to be using the term, because it comes from the particular symmetry properties of the solution of the laws of GR that you are looking at, not from those laws themselves. A "global simultaneity" or "global instant" in the sense you appear to be using those terms would be something that is built into the laws themselves. And there is no such thing in GR.

One way of illustrating that with FLRW spacetime is to choose different coordinates, such that coordinate time is no longer synchronized with the proper time of comoving observers, and the spacelike slices of constant coordinate time are no longer the same as the homogeneous and isotropic slices referred to above. This is perfectly possible; it's just inconvenient because it makes the math more complicated without any compensating gain.

 

[Matterwave]

I've often seen it said that, due to Relativity of Simultaneity, relativity makes a global instant impossible.

Not impossible, just arbitrary.

In Minkowski spacetime you can slice spacetime up in any number of ways. In fact, each inertial reference frame defines its own slicing.

A global instant is a surface that "every worldline reaching that particular value of τ" crosses once, whose events aren't even causally related, and is defined without reference to any observer.

It sounds to me like this global instant is a Cauchy Surface. Slicing spacetime up into layers of Cauchy surfaces is called a foliation. Each inextendible timelike worldline in the spacetime intersects each cauchy surface exactly once. This condition is necessary, but I don't recall whether this condition is sufficient to define a Cauchy surface. The usual definition is that the union of the future and past domains of dependence of a Cauchy surface is the entire spacetime. I don't have a reference with me to double check. Such a spacetime with a cauchy surface is called "globally hyperbolic".

FLRW spacetime is globally hyperbolic and there you may define these "global instants". So is Minkowski spacetime and so is Schwarzschild. However, how you foliate spacetime into Cauchy surfaces is not unique. You have any number of ways to do this slicing. That is the sense in which "no absolute time" exists in relativity. That's not the same as no global time.

In fact, stable causality, which is a much weaker causality condition than global hyperbolicity, already guarantees the existence of global time functions.

These global properties are examined in Wald chapter 8.

I got preempted in my response haha. Dale and Peter's responses are both excellent.

 

[OmniThoughts]

Can you provide a citation and a direct quote for this saying? The Andersson reference doesn't say this.

Do you have a source for this definition? "Global instant" is an unusual term that I have not heard of before. The Andersson reference doesn't define "global instant".

I don't see any reason that "global instants" (defined as you do) should be impossible in general relativity. You may be mixing up things from special relativity and general relativity. It is hard to tell with inaccurate citations.

It says "According to the special theory of relativity introduced by Albert Einstein, it is impossible to say in an absolute sense that two distinct events occur at the same time if those events are separated in space." in https://en.wikipedia.org/wiki/Relativity_of_simultaneity .

 

[Matterwave]

Global time and absolute time are not the same thing.

 

[OmniThoughts]

Can you provide a citation and a direct quote for this saying? The Andersson reference doesn't say this.

Do you have a source for this definition? "Global instant" is an unusual term that I have not heard of before. The Andersson reference doesn't define "global instant".

I don't see any reason that "global instants" (defined as you do) should be impossible in general relativity. You may be mixing up things from special relativity and general relativity. It is hard to tell with inaccurate citations.

The term "Global Instant" is not explicitly "defined via the Andersson reference". The Andersson reference does, I think, provide a "Global Instant" as defined in my post. I'm asking if the universe as a whole can share the same instant of cosmological time whereby that instant is regarding "spacetimes resulting from the Big-Bang" where τ is regular.

 

[Ibix]

The term "Global Instant" is not explicitly "defined via the Andersson reference". The Andersson reference does, I think, provide a "Global Instant" as defined in my post. I'm asking if the universe as a whole can share the same instant of cosmological time whereby that instant is regarding "spacetimes resulting from the Big-Bang" where τ is regular.

The motion of the matter in the universe picks out a notion of "now" in the same sense that the motion of the matter in the Earth picks out a notion of "rest" for us Earthlings. It's important and useful, but it's not an absolute and you're not required to use it.

 

[OmniThoughts]

Not in the sense you appear to mean.

In FLRW spacetimes, the spacetime does have a particular symmetry property, that it can be seen as an infinite stack of "slices"--spacelike 3-surfaces that can be viewed as "the universe at an instant of time", with each slice labeled by a unique value of "cosmological time". Each slice is homogeneous and isotropic, and there is a set of observers, called "comoving observers", whose proper time is the same as "cosmological time" (at least in the most common choice of time coordinate for "cosmological time"--but note that that's not the only possible one), and who always see the universe as homogeneous and isotropiic.

However, none of this constitutes a "global simultaneity" or defines a "global instant" in the sense you appear to be using the term, because it comes from the particular symmetry properties of the solution of the laws of GR that you are looking at, not from those laws themselves. A "global simultaneity" or "global instant" in the sense you appear to be using those terms would be something that is built into the laws themselves. And there is no such thing in GR.

One way of illustrating that with FLRW spacetime is to choose different coordinates, such that coordinate time is no longer synchronized with the proper time of comoving observers, and the spacelike slices of constant coordinate time are no longer the same as the homogeneous and isotropic slices referred to above. This is perfectly possible; it's just inconvenient because it makes the math more complicated without any compensating gain.

Since τ value is fixed by the universe's spacetime geometry, isn't no choice/choosing involved, since no observer is needed in something's value being fixed by the universe's spacetime geometry?

 

[OmniThoughts]

Global time and absolute time are not the same thing.

Thanks. I see now. I didn't realize that "an 'absolute something' is impossible" doesn't mean "a global (non-absolute) something is impossible". I think I get it now, "'something' is impossible in the absolute sense" doesn't mean "the same something is impossible in the sense of 'non-absolutely' or 'non-absolute' sense".

 

[Dale]

According to the special theory of relativity

So that is a quote about special relativity, not about general relativity. GR reduces to SR locally, but globally GR can differ from SR. Your question is a global one, so we don’t expect SR to apply.

 

[Dale]

The term "Global Instant" is not explicitly "defined via the Andersson reference".

Is it defined anywhere else? If not then I am uncomfortable using the term. You should stick with the terminology of the Anderson reference if that is your source.

I'm asking if the universe as a whole can share the same instant of cosmological time whereby that instant is regarding "spacetimes resulting from the Big-Bang" where τ is regular

Sure. That is the standard cosmological time coordinate. At every event we label the cosmological time coordinate by the age of the universe at that event. And the age of the universe is the supremum of the proper time along paths from the big bang to that event.

 

[PeterDonis]

how you foliate spacetime into Cauchy surfaces is not unique.

Actually, I think that depends on the spacetime. For Minkowski spacetime, yes, there are an infinite number of ways to do it. For Schwarzschild spacetime, it depends on which version of it you're looking at: if you just mean a spacetime with a spherically symmetric gravitating body surrounded by vacuum, yes, that works similarly to Minkowski spacetime, because, heuristically, it approaches Minkowski spacetime as $r \to \infty$, where $r$ is, heuristically, the radius from the center of the gravitating body.

But for the maximal extension of Schwarzschild spacetime (i.e., the one shown in a complete Kruskal diagram), I'm not sure--I think there might just be one unique foliation by Cauchy surfaces. And similarly for FLRW spacetime. Note that neither of these spacetimes approaches Minkowski spacetime as $r \to \infty$, even though both are spherically symmetric.

 

[PeterDonis]

Since τ value is fixed by the universe's spacetime geometry

No, it isn't. I carefully stated in my post that the "cosmological time" which is the same as proper time for comoving observers is not the only possible time coordinate that can be used. The presence of the family of comoving observers is implied by the spacetime geometry, but that's not the same as saying that you're forced by the spacetime geometry to use their proper time as your global time coordinate. You're not.

 

[Matterwave]

But for the maximal extension of Schwarzschild spacetime (i.e., the one shown in a complete Kruskal diagram), I'm not sure--I think there might just be one unique foliation by Cauchy surfaces. And similarly for FLRW spacetime. Note that neither of these spacetimes approaches Minkowski spacetime as $r \to \infty$, even though both are spherically symmetric.

Good points. I was thinking Minkowski. I should have said "not necessarily" in my post. :)

 

[OmniThoughts]

No, it isn't. I carefully stated in my post that the "cosmological time" which is the same as proper time for comoving observers is not the only possible time coordinate that can be used. The presence of the family of comoving observers is implied by the spacetime geometry, but that's not the same as saying that you're forced by the spacetime geometry to use their proper time as your global time coordinate. You're not.

τ, as defined in the 1998 paper, has a value at each event that is determined by the metric and is the same in every coordinate system, .. so that is what I mean/meant regarding the term "fixed by the geometry". I am/was not saying/implying that spacetime geometry forces use of τ as a coordinate. Nothing forces anyone to use τ as a coordinate. And other time coordinates are equally valid.

 

[PeterDonis]

τ, as defined in the 1998 paper, has a value at each event that is determined by the metric and is the same in every coordinate system

This is true, but in itself it is not enough to make it a "global time" in the sense of being part of the laws of GR. It's just a property of a particular solution of those laws.

that is what I mean/meant regarding the term "fixed by the geometry"

Ok, thanks for the clarification.

I am/was not saying/implying that spacetime geometry forces use of τ as a coordinate. Nothing forces anyone to use τ as a coordinate. And other time coordinates are equally valid.

Ok, good.

 

[Dale]

it is not enough to make it a "global time" in the sense of being part of the laws of GR

That is essentially why I objected to the terminology in post 12. The concept comes from the Andersson paper, but the terminology does not.

 

[PeroK]

Thanks. I see now.

It might be worth looking back to where all this started. Before SR it was thought that Maxwell's equations could only be valid in a single, universal frame of reference. SR enabled these equations to be valid in all (local, inertial) frames of reference. And that more generally the laws of physics should be valid in any (local, inertial) reference frame.

In GR, SR becomes a local approximation. And the advanced mathematics of GR allows the laws of physics to be expressed (in tensor form) independent of the coordinate system.

In order for cosmological, comoving time to be absolute, it would need to be the unique coordinate system where the laws of cosmological expansion were valid.

However, these laws are often expressed in a coordinate-independent way. Namely, that globally spacetime has a certain curvature. The expansion of space, in fact, is seen as a coordinate dependent interpretation.

It's ironic that "the expanding universe", which is seen in absolute terms in popular science is seen by GR experts as merely coordinate dependent. And not in fact an invariant law of physics!

 

[PeterDonis]

It's ironic that "the expanding universe", which is seen in absolute terms in popular science is seen by GR experts as merely coordinate dependent. And not in fact an invariant law of physics!

No, this is not correct. The expansion of the universe is an invariant: the expansion scalar of the congruence of comoving worldlines. It can be computed in any coordinate chart and is an invariant property of the spacetime.

 

[Dale]

the expansion scalar of the congruence of comoving worldlines

But is that scalar the same as the Hubble constant?

 

[PeterDonis]

But is that scalar the same as the Hubble constant?

Strictly speaking it's three times the Hubble constant, if you choose comoving coordinates. But its value at a given point in spacetime remains the same even if you change coordinates--whereas, at least by some definitions of "the Hubble constant", that does not.

 

[PeroK]

No, this is not correct. The expansion of the universe is an invariant: the expansion scalar of the congruence of comoving worldlines. It can be computed in any coordinate chart and is an invariant property of the spacetime.

Can the "expansion" scalar be interpreted physically or mathematically in any other way? Or, is it definitely an expansion of space over time? Are there no other valid interpretations?

 

[PeterDonis]

Can the "expansion" scalar be interpreted physically or mathematically in any other way?

The standard interpretation of the expansion scalar of any congruence of worldlines is that It's the fractional rate at which the proper distance between the worldlines increases with proper time along the worldlines. That interpretation is invariant, since it involves only invariant quantities (proper distance and proper time).

The congruence of comoving worldlines in FLRW spacetime is also the congruence that is everywhere orthogonal to the spacelike 3-surfaces that are homogeneous and isotropic, and which in standard FLRW coordinates correspond to "space" at each instant of "time". So it's common to speak as if the expansion scalar of those worldlines is a measure of "expansion of space with time". But that interpretation depends on that particular choice of "slicing" of the spacetime into "space" and "time". That choice does match up nicely with the symmetry properties of the spacetime, but it's still a choice, and other choices are possible.

So there is something of a fine line here (and to be fair, many cosmologists are not at all clear about this). There is an invariant that can be said to correspond with "the expansion of the universe", because "the universe" can be taken to refer to the spacetime itself as a geometric object, and the expansion scalar of the congruence of comoving worldlines is derived from invariant geometric properties of that spacetime. But if we call it "the expansion of space with time", strictly speaking, we're no longer talking about something invariant, because we've used the words "space" and "time" instead of "universe", and those words imply a choice of slicing which is not the only possible choice, even though that choice does match up with the same invariant geometric properties of the spacetime that we use to derive the expansion scalar.

is it definitely an expansion of space over time?

I would say no. See above.