# Question about curvature of space

1. Dec 11, 2007

### op684

I'm new to physics but very curious about it. I'm 18, and probably will also include physics as my second major in college. A topic that has always bothered me is the curvature of space. If space is curved, due to the planets and stars, then why don't the rays of the sun curve around the earth and pass it, instead of penetrating the atmosphere as they do? That's just one question I have regarding curvature of space. I'll post more very soon.

Edit: if it's due to the small size of the Earth, then with a much bigger object such as Jupiter, the rays would be deflected? Also, is there any way to measure the curvature of space around an object? Thanks.

Last edited: Dec 11, 2007
2. Dec 11, 2007

### jcsd

The curvature of spacetime is used to model gravity and gravity is an attractive force, so rather than light curving away from the Earth it should curve towards it. This is indeed what GR predicts, but the effect is very small due to the relatively small mass/desnity of the Earth.

The bigger the object the more it causes light to curve towards it. The ultimate example of this would be a (Schwarzchild) black hole which causes so much curvature that all light passing within it's event horizon will end up in the singualrity at it's centre.

Curvature can be measured by effects such as gravitational lensing.

3. Dec 11, 2007

### cesiumfrog

See Eddington's solar eclipse photographs.

4. Dec 11, 2007

### Chris Hillman

Hi, op684, welcome to PF!

You've already gotten some replies but let me weigh in with partially redundant comments:

The general theory of relativity (gtr) and allied relativistic classical field theories of gravitation (in a large class called "metric theories") treat gravitation in all or in part in terms of spacetime curvature. In this context, spatial curvature usually refers to the curvature of a spatial hyperslice. A very good introductory book which will give you some good intuition is Geroch, General Relativity from A to B, University of Chicago Press.

I don't understand what you think "should" happen (bend around and pass the Earth? as in the so-called "cloak of invisibility"?) and why, but see another good popular book by Wald, Space, Time, And Gravity: The Theory Of The Big Bang And Black Holes for the so-called light bending effect, one of the classical solar system effects which is explained by gtr.

Due to the small size of the Earth? I don't understand what you mean, but this certainly doesn't sound like a good description of light-bending as that term is used in gtr.

You probably meant to ask about the curvature of spacetime. In gtr (and allied theories), a spacetime model is a Lorentzian manifold with additional mathematical structure (e.g. a stress-energy tensor). In gtr (not in most other theories) all gravitational phenomena are understood entirely as effects of spacetime curvature. The curvature of a Lorentzian manifold is measured by a (fourth rank) tensor called the Riemann curvature tensor. The Einstein field equation EFE features the "trace-reverse" of a "trace" of the Riemann tensor, a (second rank) tensor called the Einstein tensor, on the left hand side (LHS). The EFE can be written $G^{ab} = 8 \pi \, T^{ab}$. The RHS is the stress-energy tensor which measures how much ("nongravitational") mass-energy and momentum there is at each event in the spacetime. This all gets rather technical and you probably will need an advanced undergraduate background to understand it, but now you have something to aim at over the next few years!

(For advanced students: the Gauss-Weingarten equations related the Riemann tensor of the spacetime to the Riemann tensor of a spatial hyperslice, say a slice from the hyperslicing orthogonal to a vorticity-free timelike congruence.)

5. Dec 11, 2007

### op684

Wow, Chris, thanks a lot for such a detailed response. That will definitely keep me busy for the next few years :rofl: But that's exactly what I was looking for.

Thanks to other posters too.

But to clear up my question, I've read that the reason scientists know that space (or spacetime) is curved is because light coming from distant galaxies curves around other starts/planets/etc that it comes across and continues on its path, instead of being absorbed by whatever is blocking it, and that that's how we get to know that that distant galaxy existed; something to that effect. Sorry if its confusing but I'm also confused. I guess I'll need to start from square one, and then continue on to more advanced topics such as this.

6. Dec 11, 2007

### malty

I'm not really going to reply to your question as such because my comment would only have been half as detailed as Chris's or the future comments of the other gurus here and would just be repeating others. So what am I replying for ..well merely to paraphrase bohr on QM, which I think applies to everything.

If you're not confused by it then you clearly don't understand it.

Welcome to PF, :D

7. Dec 11, 2007

### Denton

Simply put, the earth doesnt have the mass to bend light to a significant degree. Even the largest suns only fractionally bend light.

8. Dec 12, 2007

### op684

Hehe, thanks. :D

9. Dec 12, 2007

### jcsd

As I said gravity is an attractive force. We know that Newtonian gravity only causes objects to move towards each other and never causes them to move away from each other. As Newtonain garvity is a limit of GR we would expect it to produce simlair effecst which it does.

In Newtonian gravity a light ray passing by an object such as a large star, it will contiune pass it in a straight line unaffected by the star. In GR the same ray of light will have it's path altered due to the curvature of spacetime so that it passes clsoer to the star than would've been predicted by Newtonian gravity, this leads to effects such as gravitational lensing (in which the curvature of spacetime can produce a lensing effect much like the one due to classical lenses).