Questions about spacetime curvature and geometry.

[md65536]

We (humans for all of history) live in a weak gravitational field, and thus in essentially "flat" space-time, correct?

 

And the curvature of that spacetime has been constant throughout history (to the limits of our ability to measure it, at any time in history)?

 

Is it possible to observe Earth from a location in a weaker gravitational field ("flatter spacetime") and observe that what we see as "flat" appears curved from that perspective?

 

Then, would geometry appear different? Would a unit circle on Earth appear to have a circumference other than 2Pi when measured from this "flatter" perspective? Or are we basically already so perfectly flat that there would be no difference, and Pi would be the same value because it is based on "perfect flatness" or something like that???

 

Or are geometrical constants such as Pi not even dependent on something like "relative flatness"?

[ajb]

[math]\pi[/math] comes from circles and triangles etc in Euclidean space. If one is not in a Euclidean geometry then it is not necessarily true that, for example the sum of the angles of a triangle will not be equal to [math]\pi[/math]. This would also vary over the space in general.

 

So, [math]\pi[/math] does depend on the space in which the geometric figures are drawn, in the sense above.

 

Now, in general relativity we know that locally space-time looks flat, so [math]\pi[/math] has the usual geometric interpretations locally.

[md65536]

[math]\pi[/math] comes from circles and triangles etc in Euclidean space. If one is not in a Euclidean geometry then it is not necessarily true that, for example the sum of the angles of a triangle will not be equal to [math]\pi[/math]. This would also vary over the space in general.

 

So, [math]\pi[/math] does depend on the space in which the geometric figures are drawn, in the sense above.

 

Now, in general relativity we know that locally space-time looks flat, so [math]\pi[/math] has the usual geometric interpretations locally.

I see, so local space-time appears the same to everyone, regardless of gravitational field (assuming local spacetime homogeneity).

Is space-time curvature relative then? Not only does it appear flat locally, but it IS flat in all ways, locally?

 

 

 

If I'm in a "medium gravitational field" say, and I look at an area with a stronger field (eg. gravitational lensing caused by galaxies), it will appear more "closed", and the angles of a triangle will add up to more than 180 degrees, right?

 

If I look at an area with a weaker field, will it appear more "open"? Or flat? Or will all difference in curvature appear similarly, regardless of whether it's associated with a stronger or weaker gravitational field?

 

 

 

 

Then... I assume there can be an absolute lack of gravitational field, but is there an absolute flatness (or openness?) associated with that? Or is it completely relative, such that some other force or mechanism can always make space appear more open somewhere else, even if I'm experiencing no gravitational field?

Edited April 4, 2011 by md65536

[IM Egdall]

So-called spacetime curvature is the warping of space and time in the presence of matter (and energy). Imagine a place in outer space so far away from all stellar objects that gravity here is essentially zero. Here a clock runs at a certain rate. Now put this same clock in the neighborhood of a stellar object (say the Sun) and it runs slower (compared to clock in zero gravity). And the closer to the Sun, the slower the clock runs. This is time being warped by the mass/energy of the Sun.

 

A ruler in zero gravity marks out a certain distance. Now put this ruler in the neighborhood of the Sun. If placed perpendicular to the Sun, it stretches compared to its length in zero gravity. The closer it gets to the Sun, the more it stretches. This is space being warped by the mass/energy of the Sun.

 

Together this warping of time and space is called spacetime curvature. In the zero gravity location, there is no warping so no spacetime curvature. Here we say that spacetime is flat. And yes, spacetime gets more and more curved (more warping of space and time)) as you get closer to the Sun.

 

The Sun produces a relatively small amount of time and space warp, so spacetime curvature is weak (as you put it) in our solar system. But it is not really flat. There is some curvature, some warping of space and time -- this is what causes the planets to orbit the Sun.

 

Also the warping of space by the Sun affects the diameter of the Sun but not its circumference. So the ratio of the two (as measured from far away) does not equal Pi.

 

If you are trying to compare curvatures, I think the best way to do it is to look at it from the zero gravity location. Start with no curvature there and see how it changes as you get closer to the stellar object (source of curvature).

Edited April 4, 2011 by I ME

[md65536]

A ruler in zero gravity marks out a certain distance. Now put this ruler in the neighborhood of the Sun. If placed perpendicular to the Sun, it stretches compared to its length in zero gravity. The closer it gets to the Sun, the more it stretches. This is space being warped by the mass/energy of the Sun.

Is that true? I thought it only contracted.

 

Yes though: Length is really the measure that I'm interested in.

 

If you had a meter stick that is 1 m long in local spacetime, will any observer ever measure that stick as longer than 1 m, from any inertial frame?

 

 

I realize now that acceleration can allow this to happen. For example, pull on the near-end of a stick and the near end will appear to you to move before the far end does, making the stick appear longer. In this case is the stick actually longer than a meter for a moment?

 

 

[ajb]

I see, so local space-time appears the same to everyone, regardless of gravitational field (assuming local spacetime homogeneity).

Is space-time curvature relative then? Not only does it appear flat locally, but it IS flat in all ways, locally?

 

The local flatness is one way of stating the equivalence principle (just about). Mathematically this is just the fact that space-time is described by a Riemannian manifold. You can always find a neighbourhood of a point p, such that at p the metric becomes the Minkowski metric and the first partial derivatives vanish p. In essence in these special coordinates known as Riemann normal coordinates, near this point the space-time looks flat "to first order". In small enough regions the space-time flat. Einstein identifies such a choice of coordinates with local inertial frames.

 

The components of the Riemann curvature tensor do depend on the coordinates used. The fact that they are not all vanishing is not dependent on the coordinates. You need a scalar to define the curvature in an invariant way.

 

One way of doing this is

 

[math]R^{abcd}R_{abcd}[/math]

 

where [math]R^{a}_{\:\: bcd}[/math] is the Riemann curvature tensor.

[IM Egdall]

Is that true? I thought it only contracted.

 

Yes though: Length is really the measure that I'm interested in.

 

If you had a meter stick that is 1 m long in local spacetime, will any observer ever measure that stick as longer than 1 m, from any inertial frame?

 

 

I realize now that acceleration can allow this to happen. For example, pull on the near-end of a stick and the near end will appear to you to move before the far end does, making the stick appear longer. In this case is the stick actually longer than a meter for a moment?

 

 

 

Take a look at page 2-9 to 2-10 of the link below. It gives an example of space being stretched by the presence of mass/energy:

 

http://www.eftaylor.com/pub/chapter2.pdf