Bending of Space and Time


by Curious6
Tags: bending, space, time
Curious6
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#1
Aug28-04, 03:25 PM
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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?
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turbo
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Aug28-04, 03:41 PM
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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).
Hurkyl
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Aug31-04, 11:35 PM
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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)

pmb_phy
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Sep1-04, 02:03 AM
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Bending of Space and Time


Quote Quote by 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?
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.
Quote Quote by turbo-1
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.
Actually that is not what Einstein said. In fact he said
I do not agree with the idea that general relativity is geometrizing Physics or the gravitational field. The concepts of Physics have always been geometrical concepts and I cannot see why tghe gik field should be called more geometrical than f.i. the electro-magnetic field or the distance between bodies in Newtonian mechanics. The notion comes probably from the fact that the mathematical origin of the gik field is the Gauss-Riemann theory of the metrical continuum which we are wont to look at as part of geometry. I am convinced, however, that the distinction between geometrical and other kinds of fields is not logically founded.
That was in a letter from Einstein to Lincoln Barnett.
Masses curve space, so orbits can be expressed as "paths that best conserve momentum".
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
RingoKid
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Sep1-04, 04:56 AM
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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.
turbo
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Sep1-04, 06:43 AM
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Quote Quote by pmb_phy
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
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.
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Sep1-04, 08:49 AM
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Quote Quote by Curious6
How can concentrations of mass (such as the Earth) or energy bend space and time?
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.

I mean, is there any theory that states what causes space-time to be bent by these large masses?
Relativity. I'll move this topic to that forum.

Aren't space and time just abstract concepts, so how can they be bend?
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.
pmb_phy
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Sep1-04, 09:18 AM
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Quote Quote by turbo-1
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 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.
Do you have a different definition of momentum?
The momentum, p, of any body with rest mass m0 is [itex]\bold p = \gamma m_0 \bold v[/itex] where [itex]\gamma = dt/d\tau[/itex]. This quantity is not conserved for the earth orbiting the sun. Do you have a different definition?
As for my use of the geometric model of curvature, let me remind you that I said "to highly over-simplify".
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
turbo
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Sep1-04, 12:16 PM
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Quote Quote by 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.
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.
humanino
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Sep1-04, 12:21 PM
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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.
pervect
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Sep1-04, 10:55 PM
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Quote Quote by 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?
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.
Vast
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Sep1-04, 11:51 PM
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Iíve 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.

Einsteinís 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, weíll 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?
turbo
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Sep2-04, 06:37 PM
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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.

http://www.physicsforums.com/showthread.php?t=40705
Vast
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Sep2-04, 09:28 PM
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Cheers for the linked thread Turbo, very interesting read!

Though Iím 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?
pervect
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Sep2-04, 10:05 PM
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Quote Quote by turbo-1
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.
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?
da_willem
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Sep3-04, 04:09 AM
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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!
jcsd
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Sep3-04, 09:40 AM
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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
turbo
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Sep3-04, 11:56 AM
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Quote Quote by pervect
If this does make you feel better, great. If not, why not - what's missing?
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.


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