How Does General Relativity Explain Gravitational Movement?

[pro4life]

Well, to start off. I'm in my second year of physics, (honors last year, AP this year) and we haven't talked about any of Einstein's theory really yet. So, if I'm completely wrong about anything let me know!

The reason I'm posting this is... I've known about Einsteins 'spacetime' for a while and have seen the pictures that show how masses 'indent' the plane and the larger the amount, the lower a mass sits, the larger slope of indentation, so the larger the gravitational attraction. (hope I'm right so far..).

What I don't get is how do these objects move just because of a curve in the plane. Isn't this downward motion based upon the idea that gravity exists and is pulling it down? No matter how much you increase the slope, an object won't move if there isn't a force to move it (in this case, a force to move the object down the slope).

I hope somebody followed this and was able to understand! All insight is appreciated.I feel like i didn't get straight to the point here. Basically what I am asking is for example, the sun creates a gravitational slope and the Earth sits on the slope. WHY does the Earth move towards the sun just because it's on the sun's slope?

 

[romsofia]

The information you're missing is that objects travel in the straightest possible path. Since mass and energy curve spacetime, sometimes the path the object takes will be curved.

I think that's what you're asking anyway

 

[pro4life]

The information you're missing is that objects travel in the straightest possible path. Since mass and energy curve spacetime, sometimes the path the object takes will be curved.

I think that's what you're asking anyway

I edited the bottom of my post into a simple form of my question

 

[Vorde]

That analogy, pictures of which you've seen (I believe it's called the rubber-sheet analogy), is deeply flawed. It's a great way to explain what's happening, but as soon as you start you thinking about it, 99% of people arrive at the same question you're asking.

The problem here is, simply, that this is just an analogy for how it actually works. What actually happens, as said above, is that the paths through spacetime that objects take (called geodesics) are bent towards massive objects.

This analogy works because on a rubber sheet, the paths objects take are also bent by massive objects, but in the analogy you need gravity to begin with to get it to work while in the actual universe it is the properties of spacetime that cause the bending to occur.

 

[DiracPool]

I've known about Einsteins 'spacetime' for a while and have seen the pictures that show how masses 'indent' the plane and the larger the amount, the lower a mass sits, the larger slope of indentation

Check out this baby. It even has some cool "spacetime" music ;)

Basically what I am asking is for example, the sun creates a gravitational slope and the Earth sits on the slope. WHY does the Earth move towards the sun just because it's on the sun's slope?

The straitforward answer to your "quizzitation" is that objects moves along these curved lines called geodesics because gravity curves spacetime which creates these geodesics. The "force" you speak of is embodied in the degree of curvature of the geodesic, it is a mathematical construction. Nobody knows what these curves are or why they exist any further than that. And there isn't any physical manifestation to these geodesic/curves either. They're just a visual aid to assist conceptualization of the maths.

 

[A.T.]

for a while and have seen the pictures that show how masses 'indent' the plane and the larger the amount, the lower a mass sits, the larger slope of indentation, so the larger the gravitational attraction. (hope I'm right so far..).

As others said, that is a flawed analogy. It doesn't show the time dimension, which crucial. Look at these analogies instead:

http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html
http://www.adamtoons.de/physics/gravitation.swf

 

[haael]

What I don't get is how do these objects move just because of a curve in the plane.

The popular-science explanation is that every object "moves through time", whatever it means. When spacetime is curved, the time direction of an object is a little bit skewed compared to the time direction of the attracting mass. That means - "moving through time" for the object translates to "moving through space" for the attracting mass.

This analogy helped me to understand this phenomenon few years ago. For mathematical explanation, learn about geodesics, as others said.

 

[Fredrik]

The idea behind the concept of "spacetime" is that that maybe we can build a theory of physics around a set M whose points can be assigned 4-tuples of real numbers (1 time coordinate and 3 position coordinates), and whose lines can be used to represent the motion of physical objects.

A theory built up around such a set must include statements about how to interpret the mathematics as predictions about results of experiments. It would be very hard (and maybe impossible) to write down such statements about measuring devices doing arbitrary motion. For example, a measuring rod that's being accelerated or swung like a baseball bat or something, can bend, compress or stretch in ways that will make it hard or impossible to interpret the result of a length measurement. The simple solution is to write down these rules only for non-accelerating measuring devices. (A device is "non-accelerating" if a comoving accelerometer reads 0).

Because of this, the lines in M that represent non-accelerating motion are especially important. The full definition of "spacetime" must include something that singles out a set of lines. Then we can assume that those lines represent non-accelerating motion. This will be one assumption of many that together define a theory. In GR, that thing is called a "metric tensor field", or "metric". The metric singles out a set of lines called "geodesics" that in a certain mathematical sense are "straight", and we assume, as part of the definition of GR, that geodesics represent non-accelerating motion. GR can't explain why it's a good idea to do this. Experiments can confirm that it is a good idea, but the only thing that can explain why it's a good idea is another theory.

"Force" is then defined as a measure of the deviation from geodesic motion. This is the reason why objects unaffected by forces move as described by geodesics. Of course, someone who asks why objects unaffected by forces move as described by geodesics wouldn't be satisfied with this answer, because what they really want to know is why GR is a good theory. Unfortunately, the only thing that can answer that is a better theory.

The definition of GR also includes an equation called "Einstein's equation" that describes the relationship between the metric and the matter in spacetime. A different matter content would mean a different metric and therefore different geodesics. The rubber sheet analogy is supposed to illustrate this, but as Vorde said, it's very misleading. The "bowling ball" or whatever the analogy uses to change the shape of the sheet, represents the matter in the universe, how it changes the metric, and how this changes the geodesics. The analogy doesn't in any way explain why the metric depends on the matter content. As already stated, the only thing that can do that is a better theory, and the rubber sheet thing isn't even a theory.

 

[DiracPool]

"Force" is then defined as a measure of the deviation from geodesic motion. This is the reason why objects unaffected by forces move as described by geodesics.

If a meteor that is passing by the Earth follows, "passively," a geodesic created by the mass of the Earth, we would describe the motion of the asteroid as being caused by the gravitational "force" exerted on the asteriod by the Earth, would we not? That force is synonymous with the measure of the geodesic motion, not a deviation from it. I always thought that the stress energy tensor describing the mass defined the shape of the geodesic described by the Einstein tensor which, in turn, was supposed to represent the gravitational force exerted on the object.

 

[WannabeNewton]

That force is synonymous with the measure of the geodesic motion, not a deviation from it.

This is false. The geodesic equation describes the motion $x^{\mu }(\lambda )$ of a particle in free fall in space - time i.e. $\triangledown _{\xi }\xi = 0$ where $\xi^{\mu } = \frac{\mathrm{d} x^{\mu} }{\mathrm{d} \lambda }$. The equation of geodesic equation is something quite different: you consider a one - parameter family of geodesics and see how neighboring elements deviate from a given fixed element of the family by using separation vectors and the relative rate of acceleration of these separation vectors will be related to the Riemann curvature tensor. This helps relate it to force. The very idea of the geodesic as the path of a free - falling test particle in space - time is that it has no forces acting on it.

 

[SJBauer]

Per your query – “What I don't get is how do these objects move just because of a curve in the plane. Isn't this downward motion based upon the idea that gravity exists and is pulling it down? No matter how much you increase the slope, an object won't move if there isn't a force to move it (in this case, a force to move the object down the slope).”

First off, we need to be of the same understanding with reference to the model you are using. The slope of a mass’ indent in the plane of the SpaceTime field is just a graphical representation of the gravitational influence provided by an object’s positive mass. Therefore the larger the positive mass, the lower a mass sits in this model and the larger slope of its indentation (where a more vertical slope represents greater gravitational attraction). So in the model, the slope that the Earth is sitting on is merely a graphical representation for the force of gravitational attraction caused by the sun’s large positive mass.

Secondly, I would need you to qualify which motion of the objects are you asking about. If you are asking about why do the planets in the solar system revolve about the sun rather than just maintain a stationary position from the sun, then the answer is that the sun is rotating. The planets are being dragged along in the wake of the sun’s gravitational influence. Without this gravitational wake from the sun (i.e. if the sun stopped rotating), the planets would just fall into the sun. The velocity of the planet’s oval revolution about the sun allows it to sling shot past the sun and thereby maintain its distance from the sun, even though it is always attracted to the sun’s gravitational center. In other words, if the planet had a slower revolutionary velocity, the planet would run the risk of falling into the sun.

 

[Fredrik]

If a meteor that is passing by the Earth follows, "passively," a geodesic created by the mass of the Earth, we would describe the motion of the asteroid as being caused by the gravitational "force" exerted on the asteriod by the Earth, would we not?

This is mixing the terminology from two different theories. The stuff about geodesics is from GR, and the stuff about the force on the asteroid is from Newton's theory of gravity. GR does not describe gravity as a force.

I always thought that the stress energy tensor describing the mass defined the shape of the geodesic described by the Einstein tensor which, in turn, was supposed to represent the gravitational force exerted on the object.

My only objection to this part is that the sentence should end ...was supposed to represent what Newton's theory of gravity calls the gravitational force exerted on the object.