# Basic Big Bang

1. Jul 27, 2006

### 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 thats coming from millions of years ago?
Thanks.

2. Jul 27, 2006

### marcus

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 werent 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?)

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.

Last edited: Jul 27, 2006
3. Jul 27, 2006

### marcus

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==

Last edited: Jul 27, 2006
4. Jul 27, 2006

### 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?

Last edited: Jul 27, 2006
5. Jul 27, 2006

### 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.

6. Jul 27, 2006

### MeJennifer

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?

Well I understand the general reletivity 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.

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?

Last edited: Jul 27, 2006
7. Jul 27, 2006

### SpaceTiger

Staff Emeritus
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.).

8. Jul 27, 2006

### Parlyne

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 spacial 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).

9. Jul 28, 2006

### 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

Last edited: Jul 28, 2006
10. Jul 28, 2006

### Jorrie

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).

11. Jul 28, 2006

### MeJennifer

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.

Last edited: Jul 28, 2006
12. Jul 28, 2006

### Jorrie

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.
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.

13. Jul 28, 2006

### MeJennifer

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

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?

Last edited: Jul 28, 2006
14. Jul 29, 2006

### Jorrie

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.
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.

15. Jul 29, 2006

### 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 dependance 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 dependance of a(t).

16. Jul 29, 2006

### 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?

Last edited: Jul 29, 2006
17. Jul 29, 2006

### oldman

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.

18. Jul 29, 2006

### marcus

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!

====

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:

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.
===

Last edited: Jul 29, 2006
19. Jul 29, 2006

### Jorrie

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.
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.

20. Jul 29, 2006

### MeJennifer

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?

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.

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!

Last edited: Jul 29, 2006