Explaining Curved Space-Time to a Friend

[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.)

Thanks in advance!

Mark W. Ingalls

 

[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 traveling 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.

 

[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].

 

[hellfire]

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.)

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.

 

[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

 

[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

 

[pervect]

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.

 

[Gamecubesupreme]

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

 

[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

 

[hellfire]

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.

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.

 

[pervect]

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.

 

[SpaceTiger]

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

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:

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

 

[infinitetime]

The universe is finite, correct? ...

Is there is an "edge" or a "boundary" to the universe?

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...

 

[SpaceTiger]

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.

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

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

 

[infinitetime]

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

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

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...

 

[SpaceTiger]

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.

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?

 

[infinitetime]

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?

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.

 

[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

 

[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.

 

[turbo]

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.

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.

 

[Garth]

Is the Earth flat? If not then why should the spatial universe have to be flat?
Garth

 

[infinitetime]

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.

I didn't distinguish between the temporal and spatial infinities of the universe, so thanks for bringing that up. And, I agree completely with your advice and will try to follow it - always keep an open mind, because there are so many ideas, and we cannot determine which ones are absolutely correct and which ones are incorrect.

It is a very interesting subject, by the way.

 

[T10]

Like someone said above, the easiest way to think of finite space without boundaries, is earth, but let's shrink it down to make it even easier.
Say you have a bowling ball, in a complete vaccum, as in there is nothing outside the bowling ball, it just exists. Now put an ant onto the surface of this bowling ball, and ask the ant to find the edge or boundary. The ant will never be able to do so, since there is no edge/boundary to him. But is the surface area of the ball finite? Sure it is.

So there you have a finite area, without boundaries.

Just apply the same 2D rationale to our three dimensional world, and you should be able to see what your friend meant.

For a slight twist, imagine the bowling ball growing in size, faster than the ant could ever hope to run. Could the ant still make a full circle around?

 

[Garth]

Welcome to these Forums T10!

Garth

 

[pervect]

Like someone said above, the easiest way to think of finite space without boundaries, is earth, but let's shrink it down to make it even easier.
Say you have a bowling ball, in a complete vaccum, as in there is nothing outside the bowling ball, it just exists. Now put an ant onto the surface of this bowling ball, and ask the ant to find the edge or boundary. The ant will never be able to do so, since there is no edge/boundary to him. But is the surface area of the ball finite? Sure it is.

So there you have a finite area, without boundaries.

Just apply the same 2D rationale to our three dimensional world, and you should be able to see what your friend meant.

It's not impossible that his friend is right - it's just somewhat unlikely. Probably the biggest single thing I have against the original statement is it's definiteness. The original statement of the "friend" was not that the universe "could be" closed, it was that the universe "could not be other than closed". The later statement is definitely false. The universe could easily be open - not only that, it's probably more likely that it's open than it is that it's closed.

Certainly "finite" spaces like the surface of a sphere exist in mathematics (sometimes they are called compact spaces). That doesn't mean that's the actual geometry of our universe is compact ("finite").

The original poster's friend will be right if and only if the spatial hypersurfaces of homogeneity are spheres. If the hypersurfaces are either planes or saddle surfaces (spheres, planes and saddle-surfaces are all conceptually possible), the original poster's friend will be wrong.

Because the experimental evidence is centered around flat (plane-like) hyper-surfaces of sumultaneity, plus or minus, and because the "plus or minus" in this uncertanity has been steadily shrinking with time as we get better data, I think the evidence currently favors a flat geometry. Certainly that's the standard cosmology at this point, though one will still see all possibilitities routinely acknowledged as being still possible.

 

[Garth]

I think the evidence currently favors a flat geometry. Certainly that's the standard cosmology at this point, though one will still see all possibilitities routinely acknowledged as being still possible.

There is one piece of observational evidence that does not favour an open topology to the universe.
The large angle fluctuations in the CMB anistropy spectrum are missing. This lack is evident in the COBE, BALLOON and WMAP data and so seems to be a robust observation. One natural explanation for this is that the early universe was simply not big enough for these large fluctuations to exist, because it was (and is) closed, or compact.

The problem with this explanation is that the other evidence from that same data set is the universe is spatially flat (or very nearly flat) as you have said.
One resolution of this inconsistency in the analysis of the data is the fact that the WMAP angular measurements are conformally invariant, and therefore the universe is conformally flat.

What model would satisfy the requirement to be conformally flat and closed? Why, Einstein's original cylindrical static universe!

But this would require an alternative interpretation of Hubble red shift. Perhaps the data is telling us to do just that...

Garth

 

[Chronos]

The notion of the universe as a finite entity embedded in some larger entity is illogical. The same reasoning must be applied to the host as to the resident entity at some point. I say, let's cut our losses and assume the observable universe is everything we can ever possibly see. To speak of it being embedded in a more voluminous metaphysical 'space' is, in my mind, metababble for 'its turtles all the way down'.

 

[Nereid]

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

Ah, the old 'universe must be like [this] because of (my) logic' fallacy!

Have our excursions into the quantum world not taught us to be highly suspicious of the infallibility of our 'logic' (which mostly comes down to intuition)? Do any of us have particular insight into 'how the universe works', beyond what we get from applying the scientific method?

"turtles holding up turtles" theory

Discworld? or Bertrand Russell? "http://www.biologydaily.com/biology/Turtles_all_the_way_down"

 

[infinitetime]

Ah, the old 'universe must be like [this] because of (my) logic' fallacy!

Have our excursions into the quantum world not taught us to be highly suspicious of the infallibility of our 'logic' (which mostly comes down to intuition)? Do any of us have particular insight into 'how the universe works', beyond what we get from applying the scientific method?]"

That's not what I was getting at at all; I just have one opinion, and I am still very open to other opinions - I do not think my belief is infallible to all illogicality. None of us know how the universe works, so...

 

[Berislav]

pervect, I don't understand, how can something travel along a space-like geodesic?

 

[pervect]

Things don't physically travel along space-like geodesics, of course. However, both space-like and time-like geodesics are described by a function which assigns a position (vector) to an affine parameter, lambda. (i.e. we have functions x(lambda) and t(lambda) in the 2-d example I was talking about).

Reading back over the thread, I'm not quite sure why you thought I implied objects could physically travel along a space-like geodesic, so I'm not sure if this remark clears up the miscommunication.

On a related note:

The fact that a^2(t)*dx/dlambda is a constant has a physical interpretationa as a consered momentum when the geodesic is time-like, by setting lambda=tau. This exact interpretation is not strictly available when the geodesic is space-like, yet mathematically the quantity remains conserved. The straightforwards but tedious way of showing this is to write down the geodesic equations from the metric and the Christoffel symbols. This confirms the fact that this quantity is conserved, as one of the geodesic equations reduces to

d/dlambda (a^2(t)*dx/dlambda)=0.

 

[Berislav]

Reading back over the thread, I'm not quite sure why you thought I implied objects could physically travel along a space-like geodesic, so I'm not sure if this remark clears up the miscommunication.

In one of your previous posts:

It appears to me that if you follow an actual "straight line" in space (i.e you follow a space-like geodesic),

You showed in your equations that $\frac{d^2t}{d \tau^2}$ must be negative, which implies that after a sufficient amount of proper time objects following geodesics will go backward in coordinate time in a "flat" universe.
This (very interesting result) holds for every time-like geodesic trajectory, as you calculated them for $\tau$.

All in all, I'm not sure why you mentioned space-like geodesics.

 

[pervect]

In one of your previous posts:


You showed in your equations that $\frac{d^2t}{d \tau^2}$ must be negative, which implies that after a sufficient amount of proper time objects following geodesics will go backward in coordinate time in a "flat" universe.
This (very interesting result) holds for every time-like geodesic trajectory, as you calculated them for $\tau$.

Note that tau is formally equivalent to lambda, don't let my (perhaps unfortunate) choice of notation confuse you on the physical significance of the equatiions.

Note that the result you cite follows not for every geodesic, but rather for every geodesic in which dt/dtau is zero. But dt/dtau = 0 implies a space-like geodesic whenever dx/dtau > 0. Having both dt/dtau = 0 and dx/dtau = 0 does not give any interesting solutions (the solution is a point, in this case, rather than a curve).

We know that dt/dtau = 0 and dx/dtau > 0 imples a spacelike geodesic because

a^2(t) (dx/dtau)^2 - dt^2 > 0, which is a spacelike interval, not a timelike one.

 

[Berislav]

Note that the result you cite follows not for every geodesic, but rather for every geodesic in which dt/dtau is zero

But,

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

implies that t'' will be negative as a and da/dt are positive, and (x')^2 is positive even for a time-like geodesic.

Sorry if I'm being a bother.

 

[pervect]

No problemo...

The actual solution for the timelike geodesics may help illustrate what's going on

A convenient form for the time-like geodesics for the metric

(which is again ds^2 = a(t)^2 dx^2 - dt^2 for the 2d case)

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

this particular form for the geodesic is convenient becase t=0 when lambda=0

k and C are abitrary constants, H is the constant in a(t) = HT (not to be confused with the Hubble constant).

Both of the following statements are true for 0 < lambda < infinity

t'' < 0, t' > 1

and the limit as lambda-> infinity is t' = 1, t''=0

A sample plot is attached for k=0, C=2. Since the axes are not labelled, I should explain that it's a plot of the cosmological coordinate x which runs vertically versus the cosmological time t which running horizontally, for lambda (equivalent to proper time tau) varying between 0 and 5.

You can see that becase t' >= 1, the cosmological time coordinate t always increases when lambda (which in this case IS equal to the proper time tau) increases.