# Space-time question

1. Jun 1, 2005

### Mark_W_Ingalls

Anybody--

A friend of mine, whom I highly respect, made the following comment the other day:

"If you travel in one direction fast enough, you'll eventually get back to where you started."

This was so strange an 'idea' I didn't even know how to respond! After awhile I asked him how he thought of that idea, and he responded:

"Well, the universe is not infinite. And you'll never come up against a wall beyond which there is no more. So how could it be any other way? Unless the universe is just expanding too fast to do the circumnavigation."

I think he has heard something about "curved space" and he thinks space curves back on itself! Can anyone help me explain to my friend that, no matter how fast you go in a "one direction" you'll never be able to get back to your starting point? (I am an engineer, not a cosmologist.)

Mark W. Ingalls

Last edited: Jun 1, 2005
2. Jun 1, 2005

### Symbreak

I carn't actually see anything wrong with what your friend said. Clearly, if the universe is bounded in space then an observer o at a point p1 will eventually come back to p1, after travelling in the same direction.
This relies on the assumptions that 1. The universe is spatially bounded 2. The velocity of o is greater than that at which the universe expands 3. There is no significant physical boundary that makes o change direction.
Because the Universe is bounded it means it could contract - so o's journey could actually be speeded up as it approaches p1.

3. Jun 1, 2005

### Chronos

It's a theoretical possibility, but only if the universe is closed [also a possibility]. This is, however, novelty knowledge. The universe is very, very big, and will continue getting bigger for a very, very long time. It would take roughly a day short of eternity to complete the trip, and the worst part is you would have no way of knowing that you had - at least until you reached the 'big crunch' [assuming there is one in the cards].

4. Jun 1, 2005

### hellfire

Curved space is a consequence of the general theory of relativity, a theory of gravitation that Albert Einstein formulated in 1916. In this theory the local distribution of matter (in general, the local distribution of any kind of energy density and momenta) has an influence on the local distances and time intervals (the "metric" of "spacetime"). This is usually formulated saying that matter curves spacetime (a four dimensional "space" called "spacetime"). As spacetime is curved, space (three dimensional space) may be also curved.

If one makes the basic assumptions of a homogeneous and isotropic distribution of matter through the whole space of the universe (which agrees with observations on scales greater than 100 megaparsec) and one takes a specific direction for the cosmological time, then, applying the general theory of relativity, it turns out that the universe can be spatially flat, closed or open, depending on the average density of matter. You can imagine the closed universe with an analogy with a two dimensional curved space (instead of three dimensional): the surface of a sphere, in which the galaxies would be points on it. In such a case your friend would be right. Observations indicate that space is nearly flat, but this does not mean that the universe cannot be closed: it could be very big and just appear to be flat in our neighborhood.

However, space of the universe is expanding (the sphere is expanding like a baloon). If the expansion speed accelerates, then there may be objets which are never reachable, now matter how fast (but always slower than the speed of light) you go in one direction. Such a limit is called "cosmological event horizon". Observations indicate that the expansion of space is actually accelerating.

5. Jun 1, 2005

### Garth

The universe is also expanding; and thought to be accelerating in its expansion, so in fact if the universe were closed it would still be impossible to circumnavigate the universe unless you went faster than light.

Garth

6. Jun 1, 2005

### Mark_W_Ingalls

Gentlefolk--

Thank you all sincerely for your answers so far. Perhaps (OK, probably!) I am too much of an engineer to think about such things, but still...

The universe is nearly flat, correct?

The universe is finite, correct?

Then how can it be analogous to a sphere, instead of being analogous to a saddle, or a paraboloid, or a section of the surface of a sphere?

(Sure, I am stupid, but not scared to admit it...)

Mark

P.S.: Is there is an "edge" or a "boundary" to the universe? If so, and if I am at the boundary and look "out", I see nothing; but if I look "in" I see "something". How can I head "out" and get "in"? (If, if, if, ... )

m

Last edited: Jun 1, 2005
7. Jun 1, 2005

### pervect

Staff Emeritus
The universe is _probably_ spatially flat. Any spatial curvature would have to be very slight according to current data.

This means that your friend is probably wrong. He's also probably wrong about being able to circumnavigate the universe within the lifetime of the universe - but he admits that this might be the case, so it's no big deal.

However, it is indeed possible to have a universe where the spatial surfaces of constant cosmoloigcal time t are not planes but spheres. This is what's known as a "closed universe".

I've been working on a somewhat related problem in another very long thread. One conclusion that I have recently come to is that in a spatially flat but expanding universe, the path in space-time expressed by

t = constant, x = lambda, where lambda = -infinity..infinity

using "cosmological coordinates" is not really a straight line. It's a bent line, because it doesn't satisfy the geodesic equation. It appears to me that if you follow an actual "straight line" in space (i.e you follow a space-like geodesic), you will eventually wind up going backwards in cosmological time, and eventually wind up at the big bang. I've only worked this out exactly for one specific case, but I think the result is pretty general.

I would like to invinte Garth or hellfire or some of the other posters here experienced with cosmology to confirm or deny this, because it's a fairly new concept for me, though the math certainly looks good to me at this point.

8. Jun 1, 2005

### Gamecubesupreme

I don't think anything is wrong with that statement.

9. Jun 1, 2005

### Mark_W_Ingalls

Ah, gentlefolk, thank you very, very much!

As I read and re-read your very charitable posts, I remembered "The Official String Theory Website" (http://superstringtheory.com/). Of course, all of your kind responses dovetail perfectly with Dr. Schwarz's!

(I am blessed with an insatiable curiosity, but cursed with a bad memory, so sometimes I travel in non-relativistic, closed curves! )

Thanks again!

Mark

10. Jun 2, 2005

### hellfire

Sorry, but I don’t understand this. If you take a spatial hypersurface for a fixed cosmological time, a straight line within it will not bring you to the big-bang (you took a fixed time value to define the hypersurface). You are right that the straight line in space is not a comoving geodesic in spacetime as it is not comoving with expansion. May be you should elaborate.

11. Jun 2, 2005

### pervect

Staff Emeritus
OK. We have a flat FRW metric ds^2 = a(t)^2 (dx^2 + dy^2 + dz^2) - dt^2.

If we simplify the problem by setting y=z=0, restricting the problem to 2 dimensions, I calculate the geodesic equations (from the Christoffel symbols) as:

x'' + 2 [da/dt / a] x' t' = 0
t'' + a da/dt (x')^2 = 0

Here $$x = x(\tau)$$, $$t=t(\tau)$$, $$\dot{x} = \frac {dx}{d \tau}$$, $$\dot{t} = \frac {dt} {d\tau}$$

I am calling the solutions to these equations a "straight line" when the solution is spacelike.

The first point:
By this definition, x(tau) = tau, t(tau) = constant is not a straight line, as it does not satisfy the above equations.

The second point

We wish to approximate t(tau) = constant to define a specific spacelike geodesic. The closest we can come is to set dt/dtau = 0. But we see that

d^2 t / dtau^2 must be negative, since dx/dtau must be non-zero. This implies that the geodesic must start to curve. More on this later.

The third point:

If we let a(t) = H t, where H is a constant (a bit unphysical but easy to calculate) we can actually find the solution to these equaitons

The equations reduce to

x'' + 2/t x' t' = 0
t'' + H^2 t x' ^2 = 0

Solution is aided by noting that the first equation reduces to

d/dtau (a(t)^2 dx/dtau) = 0

i.e. x' = C / t^2

The solutions I'm getting are

x := k+1/2/H*(ln(H*C+lambda)-ln(H*C-lambda))
t := sqrt(H^2*C^2-lambda^2)

A sample plot is attached with C=2 and k=0. Note that dt/dtau = 0 at t=2, making this a spacelike geodesic.

I'm saying that this funny-looking (in cosmological coordiantes) curve is what is really a straight line - because it's a geodesic.

Note that d^2 t / dtau^2 is negative. The graph makes the physical significance of this fact clearer - the geodesic "bends" so that it goes backwards in time.

[add] Another way of saying this - the maximum value of the cosmological time coordinate occurs at the point where dt/dtau= 0, so that the cosmological time is less than this maximum value at other tau values. We know that it's a maximum and not a minimum because of the sign of t''.

I've heard the general statement before that "looking outwards in space is like looking backwards in time", but I never took it literaly - until I worked out the equations.

#### Attached Files:

• ###### geo.jpg
File size:
9.7 KB
Views:
63
Last edited: Jun 2, 2005
12. Jun 2, 2005

### SpaceTiger

Staff Emeritus
Careful with your use of symbols here. There's nothing technically incorrect with that, but H usually represents Hubble's constant, the fractional rate of expansion. A constant value of Hubble's constant would in fact lead to a universe whose scale factor evolved as:

$$a \propto e^{Ht}$$

13. Jun 2, 2005

### infinitetime

Well, I'm just throwing this out there, but the universe cannot just end (as a finite one would), there must be something beyong the ending, right?

The universe has to be contained in something, just like everything is contained on earth, the earth is contained in the solar system, the solar system contained in the milky way, the galaxy contained in the universe, and so on.

Therefore, the universe must be contained in something as well; and what the universe is contained in must also be contained in something else, and so on......we can call it the "containment theory" (just for kicks!). So this containment continues on and on forever, or infinitely, which indicates that the universe is infinite, I'm guessing.

I am probably wrong, and I am not sure if I make sense, but this is what I picture; I need some clearing up on this also.....

14. Jun 2, 2005

### SpaceTiger

Staff Emeritus
Sounds like the old "turtles holding up turtles" theory.

Why do you think the above must be so? Why must "containment" continue indefinitely?

15. Jun 2, 2005

### infinitetime

I know you're not criticizing, right? Just making sure, because I'm not sure what is right, no one is, so I just threw that out there.

And I believe containment must continue on indefinitely because when something is contained in something else, what is that contained in?

Ok, that's a little confusing, so let me put it this way: if the universe is contained in something, what is that contained in? It just suddenly ends?

I'm not explaining this too well, so I don't know.....

16. Jun 2, 2005

### SpaceTiger

Staff Emeritus
I understand, I'm simply trying to get you to think more about your reasoning. Try stepping back and thinking about why you feel that everything has to be contained. Is it because of your experience? Are you sure that your experiences are applicable to the universe as a whole? Are there logical reasons for such a requirement? What aspect of containment grants something its existence?

17. Jun 2, 2005

### infinitetime

Well, it is possible, anything is because no one has any real, verifiable data that I know of....

As for why I think everything has to be contained, I don't believe matter just exists on its own, floating around in emptiness (even if it was floating, it must be floating inside of something ); there has to be something every piece of matter is enclosed in or held in, if you know what I am saying.

And, I believe this belief is pertinent to the universe as a whole; the universe is matter, and that matter must be held by something, or it is "floating" inside of something.......

And maybe my reasoning is not logical, which is why I am not forcing you or anyone else to believe in it.....plus sometimes the most illogical, unlikely, or implausible ideas can be true...

Please offer me your ideas Space Tiger; do you believe the universe is finite and ends at one boundary with nothing beyond it at all? I am just wondering, not carping.

18. Jun 2, 2005

### Garth

infinitetime A finite universe does not end, at least not in the normal understanding of cosmology, a spatially finite universe is a three dimensional analogue of the two dimensional surface of the Earth; it is finite yet has no boundary. Keep on going and eventually you will end up where you started! In fact while the universe is expanding you will not be able to circumnavigate the universe and return to Earth unless you travel faster than light; but I have recently said that somewhere else!

Garth

19. Jun 2, 2005

### infinitetime

I'm sorry, I thought finite meant limited or restricted, which would infer that there is an ending, but I guess I am not familiar with the cosmological aspect of the term.

20. Jun 2, 2005

### turbo

Don't apologize for asking for clarification. While some folks seem to think they understand the concept of finiteness (and try to make an either/or argument about it), it is very difficult to define finiteness or infinity. There are degrees of infinity, and some definitions of infinity may bear upon cosmology in ways that are not well-understood. For instance, we can say the interval between 1 and 2 is only 1 in ordinal numbers. That is a pretty finite interval, and it is a concept that we expect Kindergarteners to appreciate. We can also say that the interval between 1 and 2 is infinitely divisible. In other words, we can progressively divide that interval into finer and finer intervals forever and never encounter a finite limit. This is one level of infinity. This level of infinity (in terms of our universe) might actually have a practical limit. This limit may arise as the Planck scale, below which the texture of the universe is indivisible. Mathematical ideals of infinitely divisible intervals might break down at this practical level. There are a whole lot of other definitions of infinity at various levels. There are probably an infinite number of methods to define infinities within finite intervals, before you actually have to confront the really scary infinities, like those that can obtained if the universe is spacially and temporally infinite.

If you will search on "Mandelbrot set", you will see that the field of complex numbers allows us to graph functions in the complex plane (real numbers on one axis and imaginary numbers on the other) that generate results that are infinitely complex in a very tiny section of the plane. If you have enough computing power, you can delve deeper and deeper into the set with incredible self-similarity and absolutely no duplication. This can be pretty eye-opening!

How about the big infinities? The universe could be temporally infinite (existing for all time in the past and in the future) and/or it could also be spacially infinite. Such universal models were more poplar in the 19th C and early in the 20th C, but became less fashionable after the Big Bang idea caught on. In the buttoned-down BB model, the universe began 13.7 billion years ago and every gross cosmological effect that we see now is determinalistically extrapolated back to the perfect singularity that started the universe. I personally have trouble with this concept, but lots of folks are very content with it. Study, and try to keep an open mind.