# Bending of Space and Time

1. Aug 28, 2004

### Curious6

How can concentrations of mass (such as the Earth) or energy bend space and time? I mean, is there any theory that states what causes space-time to be bent by these large masses? Aren't space and time just abstract concepts, so how can they be bend?

2. Aug 28, 2004

### turbo

To highly over-simplify, Einstein said that Newton's concept of gravity as a force acting over a distance should better be expressed in terms of geometry. Masses curve space, so orbits can be expressed as "paths that best conserve momentum". In a flat space with no massive objects nearby, that path would be a straight line, but in space surrounding a massive object like the sun, the path that best conserves momentum is going to be curved. Not necessarily round, mind you - some periodic comets have very eccentric but stable orbits (at least until they pass too close to another massive body and get perturbed out of them).

In the macro world, Einstein's concept of curved space time has had a stellar record. The biggest question facing it today, is how can it be reconciled with quantum physics (VERY small scales).

Last edited: Aug 28, 2004
3. Aug 31, 2004

### Hurkyl

Staff Emeritus
Let's start with something simpler. You're probably familiar with three dimensional Euclidean geometry. This is, of course, an abstract mathematical theory. (Though, it may be employed gainfully to model reality in many circumstances)

In Euclidean geometry, we know how to talk about things "bending"; for example, we can talk about how the surface of a sphere is curved.

Mathematically, we can generalize to a higher dimensional space; we could talk about 62-dimensional Euclidean geometry, and how a 49-dimensional surface might curve and bend in that 62-dimensional space.

Einstein's realization is that our (apparently) 4-dimensional space-time doesn't look like an example of 4-D euclidean geometry; it looks more like an example of a 4-D surface in some higher dimensional Euclidean geometry.

Alternatively, Einstein realized that, while the universe looks like a 4-D Euclidean geometry on "human" scales, it doesn't necessarily look like 4-D Euclidean geometry on large scales.

(Both of the above ways of looking at it are mathematically equivalent; it's called a "manifold"... though it's a pretty deep theorem that the latter concept can always be viewed as the former concept)

4. Sep 1, 2004

### pmb_phy

Its an analogy. The term "bend" is a term borrowed from geometry. Space and time together form an abstract entity called "spacetime." If you think of spacetime as a surface then, in general, this surface will be curved when described with the mathematics of differential geometry.
Actually that is not what Einstein said. In fact he said
That was in a letter from Einstein to Lincoln Barnett.
Orbits of free particles are geodesics in spacetime. I can't see why you'd say that they 'best conserve momentum' since such a path would be a a path for which momentum is constant and such a path is not a geodesic. Please clarify.

Pete

Last edited: Sep 1, 2004
5. Sep 1, 2004

### RingoKid

In this context, would i be way off base in comparing spacetime to water with an object floating beneath the surface to represent a concentration of mass...

...the water bends around the object.

6. Sep 1, 2004

### turbo

Maybe you can help me out, here, Pete. It is my understanding that the Earth's orbit around the Sun conserves the Earth's momentum. The orbit is stable and requires no input of force from outside to keep it going. Any deviations from the orbit would require the input of force from some external source. Do you have a different definition of momentum?

As for my use of the geometric model of curvature, let me remind you that I said "to highly over-simplify". When someone phases a question in a manner that leads you to believe that they are asking for a basic explanation of a fundamental concept, it is only fair to GIVE them a basic explanation. That's what I did.

7. Sep 1, 2004

### Phobos

Staff Emeritus
As was mentioned, "bend" may not be the proper word, but it does help you visualize it. Perhaps you can think of it as mass changing the "behavior" of spacetime.

Relativity. I'll move this topic to that forum.

Although their composition is a mystery, they are a real part of this universe and form its foundation. Things like time dilation, frame dragging, etc. can be directly measured.

8. Sep 1, 2004

### pmb_phy

The momentum of the Earth is not conserved. It is constantly changing. When there is a force on a body, such as the gravitational force of the sun on the earth, the momentum of the body is not conserved.
The momentum, p, of any body with rest mass m0 is $\bold p = \gamma m_0 \bold v$ where $\gamma = dt/d\tau$. This quantity is not conserved for the earth orbiting the sun. Do you have a different definition?
Sorry. When you said that I thought that you were referring to "Einstein said that Newton's concept of gravity as a force acting over a distance should better be expressed in terms of geometry. "

Pete

9. Sep 1, 2004

### turbo

Lets start over. First, the angular momentum of the Earth is conserved as it orbits the Sun. That's what an orbit is - the "path of least resistance" through distorted space-time. No additional force is necessary to keep the Earth in its orbit. If you Google search on "conservation of angular momentum" and the word "orbit", you'll see.

Secondly, when we talk about curved space-time (the Einstein view of gravitation), we cannot ALSO throw in the Newtonian "force of gravity" as an additional force acting on the Earth. We can model Earth's orbit either with Newtonian gravity (force acting over a distance) or with Einsteinian curved space-time. Either model alone will yield acceptable results, but we cannot talk about the Earth following a geodesic in space-time AND ALSO being pulled by the Sun's gravity. That doesn't work.

10. Sep 1, 2004

### humanino

just confusion between angular and linear momentum. You guys are refering to different (both correct) statements.

As for the interpretation of GR : everybody its own. But it is worth trying to understand Pete's interpretation.

11. Sep 1, 2004

### pervect

Staff Emeritus
What led Einstein to looking at space-time curvature was gravitational red-shift. Essentially, one has a situation in which two opposite sides of a parallelogram are measured, and found not to have the same length.

This argument was first advanced by Schild BTW.

The 4-d parallelogram is formed by two consecutive light rays moving upwards in a gravitational field. The red-shift implies that clocks further down in a gravitationall field must be ticking slowly, which means that the lorentz interval (the concept of "distance" first introduced by Special relativity, and used throughout General relativity) is different when measured at the top and bottom of this parallelogram.

Since we know that this is impossible in Euclidean geometry, Einstein was led to try non-Euclidean geometry.

This result implies that space-time must be curved, but only in a very weak sense. Nowadays, people usually think of curved-space time as having a non-zero Riemann tensor. The gravitational red-shift argument does NOT actually show that. But it does present an argument that there must be varying metric components in the space-time geometry.

12. Sep 1, 2004

### Vast

I’ve been trying to gain a better understanding of what space-time actually is, and the rubber sheet analogy seems to be misleading, well at least to me because it implies that there is some sort of medium or fabric, which is being bent.

Einstein’s actual quote "Space-time does not claim existence in its own right, but only as a structural quality of the [gravitational] field"

So if we observe a light source from a distance galaxy passing by a closer cluster of galaxies, we’ll notice that the light from the galaxy behind is bent around the nearer cluster. This of course is gravitational lensing, which is merely gravity bending light.

Is this right?

13. Sep 2, 2004

### turbo

I am not comfortable with the concept of "gravitational lensing" - invoking gravitation as the means by which light rays are refracted and distorted. It posits that gravity (an effect of space-time distortion) is the cause of lensing (another effect of space-time distortion). Please check this thread.

14. Sep 2, 2004

### Vast

Though I’m still not convinced. What I mean is light rays, which are refracted and distorted by gravity, and the apparent curvature of space-time seem to be one and the same? Am I missing something here?

15. Sep 2, 2004

### pervect

Staff Emeritus
Does the following description make you feel any happier?

Light rays, like matter, follow geodesics in space-time. Well, actually, this statement is only approximately true - it's almost true, but it's only strictly true when the energy of the piece of matter or the light ray is sufficiently low. Fortunately, this is a good enough approximation for typical applications including gravitational lensing (or planetary orbits, for that matter).

The ultimate origin of curvature in space-time is the stress-energy tensor, Tab, which generates a curvature in space-time according to Einstein's field equations, Gab = Tab. While Gab is zero in regions where Tab is zero, the general curvature tensor, Rabcd is not. More specifically, the Ricci component of R is zero when Tab=0, but the Weyl component of R is non-zero.

If this does make you feel better, great. If not, why not - what's missing?

16. Sep 3, 2004

### da_willem

This sounds like total abracadabra to me, could you please elaborate on your post. Is your question "what is missing mass?" or "what do you think about my explanation for missing mass?"?

I also don't understand why you say gravitationational interaction is instantaneous, when the General theory of relativity predicts it moves with the speed of light (not in contradiction to measurements).

Furthermore I don't understand what you mean by saying "To be a tensor, you must have something to pull against". So please enlighten me!

Last edited: Sep 3, 2004
17. Sep 3, 2004

### jcsd

John I'm not going to write out a long reply as I suspect that your posts will be deleted as they are inappropiate, anyway this is what a tensor actually is:

http://en.wikipedia.org/wiki/Tensor

18. Sep 3, 2004

### turbo

Thank you for your explanation, Pervect. What is missing, though is enough mass to cause the lensing exhibited by clusters, etc. To explain the amount of lensing actually observed (as explained in the Einsteinian model) astronomers have had to posit the existence of very massive halos of "dark matter" that are undetectable and are transparent to light and other electromagnetic radiation. There is currently a discussion on the Astronomy and Cosmology board where this is being knocked around a bit.

19. Sep 3, 2004

### pervect

Staff Emeritus
Sounds like the dark matter problem. You need it (dark matter) explain galacatic rotation curves as well, with current theeories.

20. Sep 7, 2004

### pervect

Staff Emeritus
Yes, there happens to be a theory that states what causes space-time to be bent by large masses. This theory is due to Einstein, and is called "General Relativity". The basics of this theory are Einstein's field equations:

$$G_{uv} = 8 \pi T_{uv}$$

The quantity on the left, the Einstein curvature tensor, is related to the curvature of space-time. The quantity on the right, the stress-energy tensor, can be thought of as the density of energy and momentum per unit volume in space-time.

So in Einstein's theory of General Relativity, the answer to "what causes space-time to bend" (the left side of Einstein's Field equation) is "the stress energy tensor Tab" (the right side of Einstein's field equation).

see for instance
http://archive.ncsa.uiuc.edu/Cyberia/NumRel/EinsteinEquations.html

A good but somewhat advanced online introduction can be found at

http://math.ucr.edu/home/baez/gr/gr.html

he also has a new tutorial paper that follows the same outline on the lanl preprint server.