Curvature of the Universe (II)

[andrewcheong]

This is the first thought that I had a question about, but it is highly dependent on a specific model of the universe, so I asked whether or not my model of the universe is feasible in a previous post, "Model of the Universe (I)". I have tried searching on Google for an answer, but I am not getting relevant results.

I want to know whether centripetal force/acceleration is applicable on objects with velocity moving across a curved space-time. For example, if, hypothetically, the universe is a hypersphere, and we lived on its surface-volume, and moved at velocity "v", then is there not a centripetal acceleration of (v^2)/r outward? The "r" would be the radius of the hypersphere, or the age of the universe in meters (converted from seconds in the normal-unit-conversion).

Thus, objects moving at different velocities across different curves cause different distortions and variations in the progress of time.

(Before I continue, I'd like to state a disclaimer that I have not gone past high school physics, and know nearly nothing of theoretical physics. These are all speculations, so forgive me if they seem naive and out of the question.)

Well, if everything I've said thus far is true, including the previous post on the model of the universe, I was thinking maybe relativistic effects concerning an object's speed and its time was simply a higher-dimensional incarnation of the basic concept: centripetal acceleration.

(And though this is an immature method, I am simply hoping that other effects such as the gain of relativistic mass are a secondary effect, or even, illusion, of a higher-dimensional centripetal acceleration. And I am "hoping" because I have not given much thought to these other relativistic effects as of yet, and I don't want to unless this one part concerning the distortion of time is confirmed to be at least "possible".)

Thank you for your time.

- Andrew Cheong -

 

[pervect]

I want to know whether centripetal force/acceleration is applicable on objects with velocity moving across a curved space-time. For example, if, hypothetically, the universe is a hypersphere, and we lived on its surface-volume, and moved at velocity "v", then is there not a centripetal acceleration of (v^2)/r outward? The "r" would be the radius of the hypersphere, or the age of the universe in meters (converted from seconds in the normal-unit-conversion).

Thus, objects moving at different velocities across different curves cause different distortions and variations in the progress of time.

(Before I continue, I'd like to state a disclaimer that I have not gone past high school physics, and know nearly nothing of theoretical physics. These are all speculations, so forgive me if they seem naive and out of the question.)

You'll need quite a lot of math to calculate this. Offhand I do not know the answer, though I've done similar calculations to find the geodesics for the flat-space case.

A cookbook brute-force approach that doesn't explain the theory can be found at

http://wps.aw.com/wps/media/objects/500/512494/math/pdf/christoffel.pdf

but it uses Mathematica (which I don't have). It also only gives you the differential equations which describe geodesics, it doesn't necessarily solve them for you.

A more refined approach takes advantage of the spatial symmetry that most cosmologies have to generate a conserved momentum. This means that you'll find that one of the geodesic equations actually says dP/dtau = constant, where P is a conserved momentum-like quantity.

[/quote]

Well, if everything I've said thus far is true, including the previous post on the model of the universe, I was thinking maybe relativistic effects concerning an object's speed and its time was simply a higher-dimensional incarnation of the basic concept: centripetal acceleration.

(And though this is an immature method, I am simply hoping that other effects such as the gain of relativistic mass are a secondary effect, or even, illusion, of a higher-dimensional centripetal acceleration.

"Relativistic mass" is probably nowhere near as fundamental to relativity as you think it is. See some of the discussion about it's role in special relativity in the relativity forum.

At a rough guess, you are trying to somehow exceed 'c' (or understand why it's a limit), and you don't realize that relativistic mass is not the barrier you face - the barrier you face is that no matter how many velocities you add together, if all of the velocities are less than 'c', the resulting velocity will be less than 'c'.

You add velocities using the relativisitc velocity addition formula - this comes directly from Lorentz invariance.

In any event, there isn't any need to "explain" relativistic mass increase in cosmological terms -- to the extent that this concept is actually useful, it arises straight from special relativity, is well understood, and has nothing to do with curvature of space or space-time and everything to do with the invariance of the Lorentz interval.

 

[andrewcheong]

I can't say I understand everything you said. I do know about the c limit and the addition of velocities. However, clearly, I do not know anything in this topic. I know of Mathematica, but I do not know it. I checked out your link (thank you) but I did not comprehend it at all. (I did notice while reading it though, the word "metric". This is a word that's been coming up in just about every reading I've done concerning theoretical physics. I wonder what math course I'll learn this in...) Instead of trying to figure it all out, I will ask questions in yes/no forms so that I can understand the answers, haha:

When moving with a velocity across curved space, is "centripetal acceleration" applicable due to the radius of the curve?

If so, is the acceleration vector pointed along the fourth dimension (time)?

If so, doesn't this mean that if you move along curved space, you are warping time?

With my elementary logic, I've simply assumed "yes" to all these questions. And the "warping of time" reminded me of Lorentz transformations, which made me think of this entire topic. Most likely, my thoughts are wrong, so could someone point me out in where the "no"s to the above questions are?

Thank you.

 

[pervect]

I did notice while reading it though, the word "metric". This is a word that's been coming up in just about every reading I've done concerning theoretical physics. I wonder what math course I'll learn this in...

A metric is a way of figuring out distances from coordinates.

Suppose one has two points on the Earth, specified by a lattitude and a longitude.

A metric basically allows one to convert the difference in lattitudes and longitudes into a distance.

Do you know calculus, or is your math at a pre-calculus level?

 

[andrewcheong]

I've taken Calculus I and Calculus II while in high school. We've done upto indefinite/definite integrals, etc. We did not get into integration in polar or parametric equations. We ended in some chapters concerning Taylor polynomials and expansions and such.

I've also taken Physics I and Physics II, although in the latter, we were cramming a semester's worth of college-level electromagnetism-physics into three months for the AP Examination, so I did not get a good grasp of Physics II.

I think I sort of understand metrics now.

 

[Chronos]

An entity following spacetime curvature [traveling on a null geodesic path] does not experience any centripetal force. It thinks it's in free fall. Time is really not an issue. It is bound to 'c', hence invariant in any given reference frame [i.e., your clock will keep perfect time in your reference frame, but has no obligation to remain synchronized with clocks in other inertial reference frames]. Spacetime curvature is only noticeable to observers in different reference frames.

 

[andrewcheong]

Hm, I'm not sure I understand how it is in freefall. I do understand the concept that the clock of the mover seems the same. What I'm trying to do is provide a more basic reason than exists today for why the clock of the mover and the clock of the observer ticks at different rates. There is mostly likely a flaw in my initial reasoning. I'll ask some friends and do some more research online.

Thank you.

 

[pervect]

OK, here is how geodesic motion in curved space-time works.

Firstly, let us describe mathematically the "path" an object follows in space-time. We will assign 4 coordinates to an object - think of them as x,y,z, and t. We define a path by writing 4 functions of some variable, $\tau$, which we can think of as being the 'proper time' of the object, i.e. the time read on the objects clock. We call 't' the coordinate time, and $\tau$ the proper time.

Then we have

$x = x(\tau), y=y(\tau), z=z(\tau), t=t(\tau)$

We can describe the path an object takes without any external forces acting on it by a differential equation, known as the Geodesic equation. We can write this equation compactly if we substitute x,y,z, and t as numbered subscripts of some variable. Remember that from our definition of path, all of these are functions of $\tau$ Thus, for notational purposes, we now consider

$t=u^0(\tau), x=u^1(\tau),y=u^2(\tau),z=u^3(\tau)$

The Geodesic equation then can be written compactly as

$$\frac{d^2 u^i}{d \tau^2} = \sum_{j=0}^3 \sum_{k=0}^3 \Gamma^i{}_{jk} \left( \frac{d u^j}{d \tau}\right) \left( \frac{d u^k}{d \tau} \right)$$

The $\Gamma^i{}_{jk}$ are called "Christoffel" symbols. Computing them in anything other than a cookbook fashion (i.e. understanding them in detail and their derivation) would require more math than you have, and more math than I would be able to explain in a short post. You can probably find a little more about them on the www if you look, but to understand them in detail you'll need to study differential geometry. It's probably worth noting that they can be computed if a metric is specified.

But we can draw some useful anologies, even without knowing the details of how all of these coefficients are calculated.

We note that "centrifugal force" looks like an acceleration in the x-direction that is proportional to the square of the velocity in the y-direction, and an acceleration in the y direction that is proportional to the square of the velocity in the x direction.

If we replace velocities, dy/dt, with rapidities $dy/d\tau$, we see that the Geodesic equation can create something that looks just like "centrifugal force"

This involves non-zero and equal values of two christoffel symbols, namely

$$\Gamma^x{}_{yy} \hspace{.25 in} \Gamma^y{}_{xx}$$

Similarly, coefficients like $\Gamma^x{}_{yt}$ generate forces proportional to velocity that are equivalent to coriolis forces.

Coefficients like $$\Gamma^x{}_{tt}$$ generate forces with no velocity dependence, forces like Newtonian gravity.

"Acceleration in the time direction" does not necessarily have a clear meaning, but if we interpret it as

$$\frac{d^2 t}{d \tau^2}$$, we see that any of the coefficients

$$\Gamma^t{}_{ij}$$

can cause this, with i and j being any number.