Exploring the Accelerated Expansion of the Universe

[daniel_i_l]

I read that we've recently found that the universe is expanding at an accelerated rate by observing the fact that objects that are further away from us are redshifted more and thus going faster were:
v = H*d
were H is the rate that the universe is expanding.
But if the universe is expanding faster and faster then wouldn't light comeing from further away and LONGER AGO in time be redshifted less since the more we go back in time the slower the universe is expanding? (it's getting faster over time)
How do we look at objects far away, see that they are receding faster than close ones and infer about the rate of expansion now (in acceleration, the further away an object is the faster it moves cause if for example the universe is doubling itself every X years then after that amount of time an object will have doubled it's distance from us so the further it is the more it goes in X years -Is That Right??) by light that's coming from millions of years ago?
Thanks.

 

[marcus]

I read that we've recently found that the universe is expanding at an accelerated rate by observing the fact that objects that are further away from us are redshifted more and thus going faster were:
v = H*d
were H is the rate that the universe is expanding.
But if the universe is expanding faster and faster then wouldn't light comeing from further away and LONGER AGO in time be redshifted less since the more we go back in time the slower the universe is expanding? (it's getting faster over time)
...

I think what may be puzzling you is that you assume that the cause of the observed redshift is a DOPPLER effect of the recession SPEED that a distant object HAD when it emitted the light.

that is not true.

the cosmological redshift is not a Doppler effect but instead it comes from the proportion by which space has been STRETCHED OUT while the light was in transit.

as space gets stretched out, so also the wavelengths of light get stretched.

You are worried that old light should be stretched LESS because maybe recession speeds weren't so big back when it was emitted. That is not how it works. The older light has been stretched MORE because it has been traveling longer and during that time space has expanded more.

If you would like formulas, the redshift number z is defined this way:
1 + z is the ratio of wavelengths

1+z = (wavelength now)/(wavelength then)

and cosmologists have a "scale factor" a(t) which tracks the size of the universe, like an "average distance between galaxies". If the universe had a finite radius, and they knew what it was, they could use that instead of a(t)-----but not enough is known yet, so they just have this arbitrary cosmic scale factor that tracks the expansion. It is often normalized to make a(present) = 1.

1+z = (wavelength now)/(wavelength then) = (cosmic scale factor now)/(cosmic scale factor then)

= a(when light was received)/a(when light was emitted)

$$1+z = \frac{a(t_r)}{a(t_e)}$$

this is the formula they use instead of a Doppler one, to give the redshift.

one can concoct weird coordinates that show recession speed as a real speed of something moving thru space---but they normally don't use such special coordinates. galaxies are sitting still in space (i.e. no Doppler) but their regions of space are getting farther apart (i.e. stretching wavelengths)
======================

you quote Hubble's Law
v = H*d

that means that a galaxy's recession speed TODAY (denoted v) is equal to the present value of the Hubble parameter (really should be written H₀ to denote today's value) times its distance from us TODAY.

Hubble found out this law in the 1930s, by observation.

the definition of H(t) is a'(t)/a(t)
the time derivative of the scale factor divided by the scale factor

We know that a'(t) is positive (i.e. expansion)
we know that a''(t) is positive (i.e. accelerating expansion)

Both a'(t) and a(t) are increasing with time, but this does not mean that H(t) is an increasing function of time.

Be careful about calling H(t) the "rate of expansion of the universe"
because then when you hear about accelerating expansion you might think that means that H(t) is increasing. It is not. It is currently decreasing.

H(t) is the ratio of two increasing things. It just happens that the numerator is increasing slower than the denominator. So the ratio is decreasing.

================
Later in your post you were asking about how do cosmologists KNOW that a''(t) is positive. (how do they know that the scale factor a(t) is not only increasing but is actually accelerating?)

there have been several threads about this.

Garth is good at answering this. Or SpaceT.

Garth, SpaceT or someone else may answer it, or they may give a link to an earlier thread where the Supernova Type IA observations are explained.

 

[marcus]

How do we look at objects far away, see that they are receding faster than close ones and infer about the rate of expansion now (in acceleration,..

You are basically asking how do we KNOW the expansion is accelerating.

this was found out in 1998 by observing very special type of thing
certain kind of supernova...type IA.

have to go, back later. here is something for starters
this is from
http://www.astro.ucla.edu/~wright/cosmology_faq.html#CC

==quote==
Why do we think that the expansion of the Universe is accelerating?

The evidence for an accelerating expansion comes from observations of the brightness of distant supernovae. We observe the redshift of a supernova which tells us by what the factor the Universe has expanded since the supernova exploded. This factor is (1+z), where z is the redshift. But in order to determine the expected brightness of the supernova, we need to know its distance now. If the expansion of the Universe is accelerating due to a cosmological constant, then the expansion was slower in the past, and thus the time required to expand by a given factor is longer, and the distance NOW is larger. But if the expansion is decelerating, it was faster in the past and the distance NOW is smaller. Thus for an accelerating expansion the supernovae at high redshifts will appear to be fainter than they would for a decelerating expansion because their current distances are larger. Note that these distances are all proportional to the age of the Universe [or 1/Ho], but this dependence cancels out when the brightness of a nearby supernova at z close to 0.1 is compared to a distant supernova with z close to 1.
==endquote==

 

[MeJennifer]

But can accelerated expansion can be explained with general relativity.
It seems to me that all the expansion models only expand the space dimensions and not the time dimension. How can that be consistent with GR?

Furthermore what would drive accelerated expansion as opposed to linear expansion?

 

[marcus]

In case anyone is wondering about the separate treatment of space and time in solutions of the Einstein equation. The
FRW metric used by cosmologists is a nice solution of GR equation and
this metric treats time separately from space (as many solutions of GR do)
Just for emphasis: the metric has an idea of simultaneity. one can say absolutely that two events are simultaneous----one can say when something is absolutely at rest---and so on.
The expansion of space happens in particular solutions of GENERAL relativity, not special relativity. So as time proceeds, space can expand, without any impact on the normal course of time (which is separate).

in individual solutions of the GR Einstein equation, pretty much everything one learns in special relativity as a general rule can turn out to be wrong (or to say it more softly, can turn out to have limited applicability )

It is easy to include accelerating expansion in the FRW metric. you just make a certain number positive in the Einstein equation and the solution you get will have accelerating expansion.

 

[MeJennifer]

In case anyone is wondering about the separate treatment of space and time in solutions of the Einstein equation. The
FRW metric used by cosmologists is a nice solution of GR equation and
this metric treats time separately from space (as many solutions of GR do)

Well that is nice but isn't it true that mass warps space and time? So isn't a solution where time is neither expanding or contracting a rather arbitrary solution?

The expansion of space happens in particular solutions of GENERAL relativity, not special relativity. So as time proceeds, space can expand, without any impact on the normal course of time (which is separate).

Well I understand the general relativity part. But not the time part.

My understanding is that GR describes (among other things) the volume contracting properties of mass. And if there is not enough mass to overcome some expanding momentum we cannot have contraction.
But isn't it true that the volume contracting properties do not just relate to space but also to time. Time contracts as well.
So to me it is rather odd when people claim to have an accurate model of expansion when there is no time contraction or expansion.

It is easy to include accelerating expansion in the FRW metric. you just make a certain number positive in the Einstein equation and the solution you get will have accelerating expansion.

Well in order to prevent mass from contracting we need to have some momentum force in the other direction, correct? I suppose that could be explained by some big bang phenomenon. But, if the expansion accelerates there seems to be another question coming up, by what force?

I mean, it is very nice that one can tune parameters to make it work but that is no explanation for the phenomenon correct?

 

[SpaceTiger]

Furthermore what would drive accelerated expansion as opposed to linear expansion?

In an FRW universe (one described by GR and satisfying the cosmological principle), any fluid with an equation of state $w<-\frac{1}{3}$ can drive accelerated expansion, where w is defined by:

$$P=w\rho$$

Here, P is the pressure of the fluid and $\rho$ is the density. Examples of potential drivers of accelerated expansion: cosmological constant, vacuum energy, a scalar field...

Linear expansion is what you get in an empty universe (i.e. one without matter, radiation, dark energy, etc.).

 

[Parlyne]

Well that is nice but isn't it true that mass warps space and time? So isn't a solution where time is neither expanding or contracting a rather arbitrary solution?

In general, yes, mass (and energy and momentum, etc.) warps space and time. However, cosmological models are not intended to be general solutions to the Einstein equation. They are solutions which account for the observation that, on large scales, the universe is homogeneous (looks the same everywhere) and isotropic (looks the same no matter which direction we look in). When you solve the Einstein equation with these constraints, you find that the geometry of spacetime looks like:

$$ds^2 = -dt^2 + a^2(t) \left (\frac{dr^2}{1-kr} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right)$$

where k describes the character of the spatial geometry (k=0 for flat, or Euclidean, space, k=1 for spherical geometry, and k=-1 for hyperbolic geometry) and a(t) is the scale parameter, which is constrained in two ways. First:

$$\left (\frac{da}{dt} \right )^2 - \frac{8 \pi \rho}{3} a^2 = -k$$

which is called the Friedman equation. And, second:

$$\frac{2}{a} \frac{d^2 a}{dt^2} = -8 \pi \left (p + \frac{\rho}{3} \right )$$.

In both equations, $$\rho$$ is the total energy density of the universe and p is the total pressure (both of which are assumed to be constant in space because of the homogeneity and isotropy).

 

[Garth]

Good answer Parlyne, just a point of clarification: the Robertson-Walker metric

$$ds^2 = -c^2 dt^2 + a^2(t) \left (\frac{dr^2}{1-kr} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right)$$

doesn't have to be derived from Einstein's field equation; it is actually the isotropic and homogeneous (maximally symmetric space) metric of any gravitational theory.

The Friedmann equations are derived from Einstein's FE and they determine the a(t) and k that appear in the above.

Garth

 

[Jorrie]

Well that is nice but isn't it true that mass warps space and time? So isn't a solution where time is neither expanding or contracting a rather arbitrary solution?

Something that cleared up a similar confusion for me was this realization (which the experts normally don't even mention ): in a homogeneous and isotropic universe of infinite size, there is no relative time dilation from point to point on the large scale. All points are equal and registering 'cosmological time'. That is why the usual metrics of GR do not apply. And this universe doesn’t need to be infinite, just large enough (many times the size of the observable universe, whatever that may mean).

 

[MeJennifer]

Something that cleared up a similar confusion for me was this realization (which the experts normally don't even mention ): in a homogeneous and isotropic universe of infinite size, there is no relative time dilation from point to point on the large scale. All points are equal and registering 'cosmological time'. That is why the usual metrics of GR do not apply. And this universe doesn’t need to be infinite, just large enough (many times the size of the observable universe, whatever that may mean).

Well on face of it that argument appears wrong.
Is seems that relativistic time dilation is used as an argument to discount the effects of gravitational time dilation.

The amount of space contraction or expansion directly relates to the amount of matter in the universe (mass, energy, basically everything that warps space-time). But the same applies to the amount of time contraction or expansion.

 

[Jorrie]

Well on face of it that argument appears wrong.
Is seems that relativistic time dilation is used as an argument to discount the effects of gravitational time dilation.

There is no 'relativistic time dilation' (taken to mean 'velocity time dilation') in the standard cosmological model. The galaxies or whatever structures, are assumed to be at rest in space. Think about the outdated 'inflating balloon' analogy. The galaxies are not moving in space – space is stretching and therefore the galaxies move apart.

The amount of space contraction or expansion directly relates to the amount of matter in the universe (mass, energy, basically everything that warps space-time). But the same applies to the amount of time contraction or expansion.

I think you are confusing the gravitational time dilation and equivalent gravitational 'space contraction' outside of a large mass (i.e., non-homogeneous space) with the homogeneous space of the standard cosmological model. Again, think about the inflating balloon analogy: if matter were uniformly distributed around the balloon, where on the surface would you find a place that has denser matter than any other place? Where would space-time be 'warped' more than any other spot on the balloon?

Be aware that translating GR straight into cosmology is full of pitfalls! IMO, the cosmological solutions to Einstein’s field equations are special cases.

 

[MeJennifer]

There is no 'relativistic time dilation' (taken to mean 'velocity time dilation') in the standard cosmological model.

Exactly, that is what I was saying. But of course that can be no argument against gravitational time dilation.

I think you are confusing the gravitational time dilation and equivalent gravitational 'space contraction' outside of a large mass (i.e., non-homogeneous space) with the homogeneous space of the standard cosmological model.

Are you suggesting that the total mass in the universe is not a factor in whether the universe expands or not?
Perhaps I don't understand what you mean. What am I confused about?

 

[Jorrie]

There is no 'relativistic time dilation' (taken to mean 'velocity time dilation') in the standard cosmological model.

Exactly, that is what I was saying. But of course that can be no argument against gravitational time dilation.

Before you reply, do yourself a favor and read the full pdf from http://arxiv.org/abs/astro-ph/0305179 (Lineweaver). Its short and not too technical, but it may answer some of your questions and make the disscussions in this forum more fruitful for you.

Are you suggesting that the total mass in the universe is not a factor in whether the universe expands or not?
Perhaps I don't understand what you mean. What am I confused about?

A completely massless universe with cosmological constant will still expand! It is called the de Sitter universe. See http://en.wikipedia.org/wiki/De_Sitter_universe. The mass of the universe obviously influences the expansion rate, but it is not a prerequisite.

As to what confuses you, from your questions and statements, I still think you read the ‘contraction of space and time’ around a black hole (say) as equivalent to (or the reverse of) cosmological expansion. They are not equivalent - they are different solutions to Einstein’s field equations. Look at the metric in the Lineweaver document referenced above.

I'm not a cosmology mentor, so maybe I'm confusing you! Garth, SpaceT and others are much better at teaching. But I am sure they will also require you to read some technical articles on your own.

 

[Parlyne]

If you want to talk about gravitation time dilation, what you need to realize is that it can only occur when the effects of gravity are different from place to place. This is necessary becuase time dilation is inherently a comparison between two different clocks. In Schwarzschild geometry, it is true that a clock farther away from the source of gravity will run faster than one closer to the source. However, in cosmological models, the effects of gravity have no dependence on position. Thus, there can be no gravitational time dilation. Let me show you how this works.

The first thing to realize is that the time measured along a path, S, will be given by:

$$\tau = \int_{S} \sqrt{-ds^2}$$

In Schwarzschild geometry, the line element is given by:

$$ds^2 = -\left (1-\frac{2M}{r}\right )dt^2 + \left (1-\frac{2M}{r}\right )^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

where all quantities are written in units where $$G = c = 1$$.

If we consider stationary objects, $$dr = d\theta = d\phi = 0$$, which is to say that the position coordinates aren't changing. Once we've done this, it's easy to see that the time measured by a stationary observer at position $$(r,\theta,\phi)$$ will be:

$$\tau = \sqrt{1-\frac{2M}{r}}t$$

Here, t is the coordinate time, which is the same as time measured an infinite distance away. We can see from this that the closer an observer is to the gravitating body, the shorter a period of time they experience during a length, t, of coordinate time.

In the cosmological models, on the other hand, the time experienced by any stationary observer will simply be $$\tau = t$$, no matter where they are. This can be interpreted by recognizing that, no matter where you are in a homogeneous, isotropic universe, you are always exactly the same distance from any given amount of matter.

Now, if we wanted to, we could define a different coordinate time. For example, we could define T, where $$dT = \frac{dt}{a(t)}$$. Then, the line element would look something like:

$$ds^2 = a^2(T)\left (-dT^2 + \frac{dr^2}{1-kr} + r^2(d\theta^2 + \sin^2\theta d\phi^2)\right )$$

However, this still wouldn't change the fact that all stationary observers see time passing at the same rate and it makes the dynamical equations, which still have exactly the same meaning, more complicated.

I should point out that the special relativistic sort of time dilation still exists here, albeit in a slightly more complicated form. Consider, for example, an observer moving with $$r = vt$$. And, just to simplify things a little, let's assume that $$k = 0$$.

In this case, we find that $$\tau = \int_{0}^{t} \sqrt{dt'^2 - a^2(t')dr^2}$$. Or, to put it more simply, $$\tau = \int_{0}^{t}\sqrt{1-a^2(t')v^2}dt$$. This, of course, can't be integrated without knowing the form of a(t). However, we can see that the structure of this looks like an adaptation of the special relativistic formula. That said, comparing times between two different moving observers is no longer so simple, specifically because of the time dependence of a(t).

 

[MeJennifer]

So Jorrie, Parlyne, are you basically telling me that in GR there is no such thing as expansion or contraction of time similar to expansion or contraction of space?

So space-time does not expand or contract it is just the space part and that is all in accord with GR?

 

[oldman]

But can accelerated expansion can be explained with general relativity.
It seems to me that all the expansion models only expand the space dimensions and not the time dimension. How can that be consistent with GR?

You might look at my post #36 in the thread "Is the Universe really expanding" which brings in possible changes to the time dimension and the interpretation of expansion.

 

[marcus]

So Jorrie, Parlyne,...

So space-time does not expand or contract it is just the space part and that is all in accord with GR?

That's always been my take on it too! I didn't realize there was any confusion about it. Yes indeed JUST THE SPACE PART and yes indeed ALL IN ACCORD WITH GR. Congratulations to Jorrie and Parlyne for maintaining and conveying this basic idea!

Probably unnecessary, but I will add my own song and dance routine about this to what has already been said
====

Of course there is a local gravitational effect on the passage of time that you get by the locally inhomogeneous distribution of matter. The lumpiness of matter means that a clock at one place may be deeper in the gravitational field than a clock at another place---the deeper clock runs slower. But that is not an expansion or contraction of SPACETIME.

I want to re-iterate what you said because I think it is progress in the discussion:

So space-time does not expand or contract

In classic GR, spacetime gets equipped with a metric which is a solution of the main equation. Thus equipped with geometry, spacetime is a static eternal thing which never changes. It cannot expand or contract. It is one possible solution to the equation, describing the geometry of the universe thru all forever from beginning to end.

A fixed crystalline history of the world.

What is meaningful is to SLICE IT UP INTO SPATIAL SLICES, and then compare successive slices. As you probably know this is called a "foliation"-----related to the Latin word for LEAF and to the word "folio" (a pile of leaves bound to form a BOOK).

A book is a good image of spacetime foliation. Then you can look at successive spatial pages of the book, numbered by moments in time, and you can ask DO THE LETTERS GET FARTHER AND FARTHER APART.

If the successive pages of the book have the letters more and more spread out, that is what one means by space expanding.

When people talk about that, it has never (in all since Einstein's first GR paper in 1915 AFAIK) ever been anyone's intention to suggest that space expanding signified "spacetime expanding"
Like Jorrie and Parlyne (and I think now you) are saying IT IS ONLY THE SPACE PART THAT EXPANDS.
I should stress that I'm not talking about the local effect of inhomogeneities----clocks running at different rates---the context here is what people mean by the global expansion of space: Hubble law stuff.

I guess you could, as a "clever idea" (actually a bad idea), do a logical maneuver where you describe the space expanding process by saying that TIME IS CONTRACTING and then rescale everything so that it would give the same picture. That would merely amount to using different clocks and rulers from everybody else and still come out with the same qualitative result. The rules of physics usually allow for some playing around with "goofy units" as long as you are consistent and explicit about what you are doing. But in this case, saying " space does not really expand, what really happens is time contracts" would have no practical purpose and might confuse others.
===

 

[Jorrie]

So Jorrie, Parlyne, are you basically telling me that in GR there is no such thing as expansion or contraction of time similar to expansion or contraction of space?

This is not at all what we told you about GR! This is what we told you about a specific solution in GR that is used in physical cosmology.

So space-time does not expand or contract it is just the space part and that is all in accord with GR?

I say again what I tried to say before: it appears as if you are confusing the Schwarzschild solution (non-homogeneous space) with the Friedmann-Lemaître-Robertson-Walker (FLRW) solution (homogeneous space) of the EFEs. On the large scales, the FLRW metric holds. When you go to smaller scales, one finds embedded into more localized space the metrics of Schwarzschild, Kerr and others, depending on the case in hand. These describe the spacetime curvatures caused by local lumps of matter. The FLRW metric has space curvature but no spacetime curvature.

And BTW, one should rather not talk about space and time that contracts/expands in GR, especially when referring to local effects. Rather stick to spacetime curvature in the presence of lumps of matter and you stay out of trouble. E.g., on the large scales, there is no spacetime curvature, only (possibly) spatial curvature (apparently, space is flat anyway). But, as observation tells us, this flat space does expand on the large scales. So one should not equate spacetime curvature to spatial expansion.

 

[MeJennifer]

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 seperately from time. Or am I completely wrong about that?

In classic GR, spacetime gets equipped with a metric which is a solution of the main equation. Thus equipped with geometry, spacetime is a static eternal thing which never changes. It cannot expand or contract.

Really, so then how do you for instance describe a closed universe (where K > 0)?
And when a closed universe contracts, space-time remains static?

It seems to me that in GR we always have to consider space-time. A 3D space slice is simply an observer dependent view on it.

When people talk about that, it has never (in all since Einstein's first GR paper in 1915 AFAIK) ever been anyone's intention to suggest that space expanding signified "spacetime expanding" or "time expanding".
Like Jorrie and Parlyne (and I think now you) are saying IT IS ONLY THE SPACE PART THAT EXPANDS.

Well, surely I am completely wrong since you guys seem to understand this much better than I do.
But it is in direct contradiction with my understanding of GR.

In GR mass curves space-time (that is space and time). Whether a universe is open or closed simply depends on the amount of mass available. If there is too much mass the universe contracts and that means, by my understanding of GR, that space-time contracts, thus both space and time. The time-reversed scenario should obviously do the opposite.
So that is why I question the validity of a cosmological model that discounts this idea.

So it seems I have to study GR better to understand how space-time remain static and only the 3D slices expand.
To me it is like complete nonsense, a sort of simplification that runs completely counter to the intricate mixture of space and time in GR.

Thanks for the explanations, I do appreciate it, but it seems my understanding of GR is seriously flawed!

 

[marcus]

... The FLRW metric has space curvature but no spacetime curvature.
...

Hi Jorrie,
I think the FRW metric can be understood as sort of a FORMAT which can implement different geometries depending on the two parameters you plug in: the k and the a(t).
The k is a spatial curvature parameter which can be chosen equal to -1 , 0 , +1.
The a(t) is the spatial scale factor.
The way you get a version of the FRW that satisfies Einstein's equation is you pick a value of k and you solve Friedmann's equation for the scale factor a(t).

In most of the cosmology papers I've seen over the past 5 years, if they have a favorite for k they usually pick k=0, the spatially flat case. It seems the universe is either perfectly spatially flat, or so close that it is a very good approximation to assume it spatially flat.

In principle, I guess, working cosmologists are open to all possibilities---and a few may take the k = +1 positive curvature case seriously. But overwhelmingly I have seen them assume k = 0 spatial flat. It is the quickest simplest way to get a good fit to the observational data.


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

 

[marcus]

And when a closed universe contracts, space-time remains static?

yes,

a given classical spacetime is the whole story from beginning to end.
it does not change. it does not evolve.
in the example you are considering the spacetime has a bigbang at one end and a bigcrunch at the other end.
it just sits there, statically, like a history book.

BTW a closed universe does not necessarily ever contract---it doesn't necessarily go crunch.

 

[marcus]

... how do you for instance describe a closed universe (where K <0)?

in the notation I am used to, a closed Friedmann universe is one where K>0
in fact one can normalize it so that K=+1

Maybe people have different conventions and in your book the signs are different so you would use K= -1 for the closed case.
But I still find it a little disturbing to see you refer to a closed universe and say "where K < 0".
Maybe it is just that we have different conventions.

I think it is important for you to realize that a closed universe does not necessarily have a big crunch.
Books written before 1998 typically assume that closed implies crunch.
Then there was the revolution in cosmology in 1998, because of the Supernova IA observations.
Evidence of accelerated expansion was observed. After 1998 it was no longer believed that closed implies crunch.

In fact the data is consistent with our universe being closed and also being destined to expand indefinitely.

 

[MeJennifer]

yes,

a given classical spacetime is the whole story from beginning to end.
it does not change. it does not evolve.
in the example you are considering the spacetime has a bigbang at one end and a bigcrunch at the other end.
it just sits there, statically, like a history book.

In a close universe we can clearly "see" the 4D closed manifold get smaller during a crunch. It is not static. If k=0 then it is static and if k>0 it is like a "saddle" that is open

And you are correct, I place a typo, k>0 for a close universe.

 

[marcus]

In a close universe we can clearly "see" the 4D closed manifold get smaller during a crunch. It is not static...

I think I will let someone else have a turn at answering you now.
I only want you to be sure of two things:

In a closed universe THERE DOES NOT HAVE TO BE A CRUNCH
because with positive Lambda you can just keep on expanding (in the k>0 case). So we can NOT clearly always "see" a crunch.
In a closed universe there might not be a crunch. It is something to get straight on since it is already 8 years since the 1998 revolution

the other thing is that a classical 4D spacetime manifold is static (as I personally use the word static.)

In a way it is like the diagram in a physics book showing the parabolic arc of a projectile.
the parabola tells the whole story. it does not move or change. it just sits there.
so I say that this picture of a parabola is static.

You can say that a picture of a parabola is NOT static. then from my viewpoint you are bending the word. which is OK but doesn't help me understand you.

BTW JUST AS A SIDECOMMENT: in the QUANTUM version of GR----say the spinfoam sum-over-histories or spacetime "path integral" approach----there is a whole BLUR of different spacetimes. a whole mishmash of possible geometric evolutions from an initial to a final geometry are considered.
so it is important to me to make the distinction between a
CLASSIC spacetime which is a single story: clear, crisply focused, predetermined.

versus the quantum spacetime which is an indeterministic fuzzy multitude of stories depicting how the world's geometry evolved.

quantum GR is an unfinished theory people are working on (you may be familiar, anyway it is not on topic)

 

[MeJennifer]

In a way it is like the diagram in a physics book showing the parabolic arc of a projectile.
the parabola tells the whole story. it does not move or change. it just sits there.
so I say that this picture of a parabola is static.

You can say that a picture of a parabola is NOT static. then from my viewpoint you are bending the word. which is OK but doesn't help me understand you.

Sorry but that is not what space-time is.
Space-time is a flat, open or closed manifold that evolves.

For instance at any point in it's evolution we can consider it's global curvature, both of space and time. This curvature can change during it's evolution.

 

[Garth]

Sorry but that is not what space-time is.
Space-time is a flat, open or closed manifold that evolves.

For instance at any point in it's evolution we can consider it's global curvature, both of space and time. This curvature can change during it's evolution.

"Sorry", but the word "evolves" implies process through time. However, if you are considering space-time, then time has already been accounted for in the manifold under consideration.

Space-time by definition has to be static, there is and can be no evolution of space-time as a whole. Individual foliations of space-time - space-like slices through space-time may, and do, indeed change as a function of the time parameter, that is from one coordinate time to another.

However such foliations, the division of space from time that any temporal observer experiences, is frame dependent. In the cosmological solution it is only because we select a homogeneous and isotropic space-like slice that we can talk about the universe's evolution.

Space-time does suffer curvature, which is indicated by the components of the Riemannian not all being zero, and this results in such homogeneous and isotropic space-like foliations being spherical, flat or hyperbolic, however the use of the word 'curvature' applied to time is a bit of a mis-nomer. In the local Schwarzschild case it is revealed as time dilation.

Garth

 

[kmarinas86]

Sorry but that is not what space-time is.
Space-time is a flat, open or closed manifold that evolves.

For instance at any point in it's evolution we can consider it's global curvature, both of space and time. This curvature can change during it's evolution.

True, however, supposed dark energy may have some way to "brace" the grid of space-time into a more Euclidean form while concurrently forcing it to accelerate outwards in a uniform fashion (think of an expanding cube). In otherwords, the acceleration is supposedly not due to curvature of global spacetime but by the forces that dark energy has on spacetime, as if there was a kind of energy that cause space to expand. This implies sort of energy that has the ability to push objects away from each other. But is this really a force as in kg*m/s^2, or does this involve a new abstraction? After all it makes no sense that a force could be exerted on something posessing no mass (i.e. the space-time fabric). Dark energy, if it really exists, its something of an "anti-friction property" with negative pressure content. Space-time possesses a coordinate system, and so does the magnetic field. Space-time expands (i.e. inflation happens), allegedly, with no cause or stimulus allowing it to happen, whereas a magnetic field can only expand if the medium the magnetic field lines travel through is becoming more permeable. Expansion occurs more easily when there is less resistance. The theme we see in the views of distant galaxies show more resistance than we see in the view of closer galaxies (given the primordial state of the early universe). In a homegenous and isotropic universe, its hard to see how we would return to that "primordial state" consisting mainly of hydrogen, since that implies a decaying of matter to lower forms with increasing resistance due to density levels returning to that of the primordial density.

 

[MeJennifer]

"Sorry" but "evolves" is a word that implies process through time. However, if you are considering space-time, then time has already been accounted for in the manifold under consideration.

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. But of course from observer frames simultanuity is different, theirs is the so called "3D plane of simultanuity". But space-time evolves and during this evolution space and time gets warped and unwarped.

Space-time does suffer curvature, which is indicated by the components of the Riemannian not all being zero, and this results in such homogeneous and isotropic space-like foliations being spherical, flat or hyperbolic, however the use of the word 'curvature' applied to time is a bit of a mis-nomer.

Really, feel free to explain why.
Time warps just like space does.

In the local Schwarzschild case it is revealed as time dilation.

In the case of a black hole time simply gets warped to infinity, which actually means that such an event has no future extensions in time (also also space of course) anymore.

In other words, the acceleration is supposedly not due to curvature of global spacetime but by the forces that dark energy has on spacetime, as if there was a kind of energy that cause space to expand.

Sure that is all very well possible, I am not discounting that at all.
But at least, when you wish to model that and claim that it is yet another "fitting Einstein equation", then don't simply ignore the time expansion or provide some theory why time is not expanding as well. I mean anybody can add some constant or some scalar and then say "see, it works with GR", but what you are really doing is ignoring GR, ignoring the intricate mixture of space and time and treating space pretty much like some pre-Copernican model.

 

[Jorrie]

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!

 

[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 seperately 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.
==========
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.

==========
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.
===========

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!

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

Agreed.

==========
…. 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.