Exploring Einstein's Theory: Gravity and the Curvature of Space

[Line]

Einstein made this idea that objects distort space. The bigger the opject the bigger the distortion.

If an object came along it would bend space like a marble on a blanket. And other near abject would fall intot he curvature and this was gravity. Although I revere EInstein I think this is absurd. If objects curve space what making them fall in? There has to be a force pulling them down into the depression. That and space is 3 dimensional. Blankets and any other objects with curves,canyons, or depressions in the are 2 dimensional.

 

[Sojourner01]

It's a bad analogy - one Einstein himself, as far as I'm aware, didn't use.

I'm not very expert on relativity - particularly general relativity, but as I understand it, mass distorts spacetime so that a moving object's concept of a straight line a la Newtonian mechanics isn't the same as the idea of a straight line held by an observer. The object thinks it's traveling in a straight line through spacetime, but we see it as being deflected toward the massive body. The point of relativity is that each is equally valid.

 

[LURCH]

I believe that the two-dimensional nature of a flat surface is rather the entire point of the analogy. The flat, 2-D surface is curved in a third direction that is niether "Back&Forth" nor "Side-to-Side"; the direction of "Up-and Down". Objects passing by either "Behind-or-in-Front" or "Left-or-Right" of the depression will change course, following a curved path in 2-D space because of the distortion of the environment in this third direction.

Likewise, an object with mass bends 3-D space in a fourth direction that is not "Back&Forth" or "Side-to-Side" or[i/] "Up&Down". Objects passing by "behind-or-in-front" of, "Left-or-Right"of, or "Above-or-Below" this distortion will change course, following a curved path in 3-D space because of a curve in a Fourth direction.

As for whether or not Einstien used the analogy of a blanket or a rubber sheet; I've no idea. But he did say that gravity curves or bends spacetime, and the analogy is very usefull and accurate.

 

[kvantti]

The 2D analogy requires an external force to curve the rubber sheet and no such force is presenet in general relativity. The "curvature" only defines the geometry of spacetime and objects travel along the straightest paths thru spacetime, as Sojourner01 pointed out.

 

[pervect]

The simplestic analogy is indeed too simplistic. A more accurate analogy describes how "geodesic deviation" causes two worldlines, worldlines which are as straight as possible in a curved space-time (i.e. geodesics), to appear to accelerate twoards one another.

Consider two explorers, starating out at the north pole, and walking towards the south pole, along a line of longitude. A line of longitude is a "great circle", which is a geodesic on the sphere.

They will initially be at the same point. They will then move away from each other, reaching a maximum distance at the equator, after which they will rejoin at the south pole.

One imagines that this physical example of motion in space is, instead, a model of the worldlines of two particles in space-time. The lattitude in this problem becomes a time coordinate.

One then sees two observers experieicning time, observers who are intially moving away from each other. These observers, each of which is following a geodesic path (as close to a "straight line" as is possible in a curved space-time) appear to accelerate towards each other, because their relative departure velocity slows down, becomes zero "at the equator", and then becomes negative as they start to rejoin and approach each other.

This is what is meant by curved space-time. The "time" part of the curvature is very important, in fact under many circumstances (but not all) the spatial curvature is not too important.

 

[Line]

Still confused. You can't distor one part of space without distorting all of it. This may work with rubber or a blanket in 2D but if you try to distort one part of a rubber block or anything 3 dimensional you will tear it. You can't cause a 3 dimensional distortion in something 3D. You might be able to bend it in one direstion or the other, but you have to bend the whole block. Besides space technically doesn't curve. A curve is the curving of an abject or geometric figure in space. If it has to be in space then it is impossible for there to be a technical curvature in space.

Now however if we could go to a higher dimension and watch space curve it woul d curve from our point of view. Technically no matter how much our space curves it would be as straight as normal to us inside of it.

 

[Sojourner01]

But he did say that gravity curves or bends spacetime, and the analogy is very usefull and accurate.

It's good when you understand precisely what it's trying to say - as you described. However, most students don't understand this - and evidently this causes confusion.

Line, You're still getting hung up on the analogy. The point of relativity is that there is a 'fourth dimension' for 3d space to curve in. As a result, what we're discussing is not curvature in space, but curvature in spacetime, which is not the same thing.

Ultimately though, all this language is shorthand for the correct and functional mathematics which describe general relativity - which I imagine we all wish could speak for themselves.

 

[russ_watters]

I don't think it's a bad analogy, it's just that you have to make sure you understand that no analogy, no matter how good, can be perfect. It is only an analogy.

Just like with the expansion of space, it is impossible for a person to visualize something happening in more than 3 dimensions because we can only see three dimensions. Translating the concept into something with fewer dimensions so we can visualize it can be very helpful.

Still, I think there may be a way to visualize this idea in real-life terms. Sometimes when you watch the sun or moon rise or set, you see mulitple images. And even when you don't see multiple images, the image is always in a different place than where the object really is. This is due to refraction. The air has different densities and temperatures, causing complex refraction. Now imagine a similar thing happening in outer space. Its just that instead of the air bending light, you have the curvature of space itself bending light. But it works similar to density variations -imagine space getting 'compressed' around a large object. Light passing the object follows the curved path of the compression. The Hubble telescope has identified many instances of gravitational lensing.

 

[kvantti]

You can easily adapt to the "gravity as curvature of spacetime" - point of view if you can just think a little bit differently:

If you throw an object at an 45 degree angle and watch it fall, you'll see that it moves along a curved path. At this point, most people would think "the object moves along a curved path in 3D space because gravity pulls it towards earth." This is the Newtonian point of view.

But a person who had just studied general relativity would think "the object moves along a straight path in a curved 4D spacetime because Earth curves the spacetime around it." The 3D projection of this motion is seen as accelerated motion towards the gravitating mass.

So, instead of thinking that gravity "pulls" objects hence causing them to travel along curved paths in 3D space, think that the objects travel along the straightest paths in curved 4D spacetime.

Oh and don't try to "visualize" how 4D spacetime "bends"; you can't. A human being isn't able to visualize in four dimensions, since we only have three dimensional models in our brains.

 

[Line]

1st of all I think it shou;dn't matter the curving. A portion of space can bend twist and turn. You won't notice anything as long as ou bend twist and turn in it. Kind of like words on a page. It may link wrinkled and curled up to us, but to the words if they were living it seems the same as always.
The spacing within the page hasn't changed. So in order for some change to happen in an object some space has to disappear or compress to the people within the universe. Oherwise it's like the words on the page.

 

[MeJennifer]

The simplestic analogy is indeed too simplistic. A more accurate analogy describes how "geodesic deviation" causes two worldlines, worldlines which are as straight as possible in a curved space-time (i.e. geodesics), to appear to accelerate twoards one another.

Correct, but I think the main case is here about the gravity, e.g. the "sucking" action.
Due to curvature or not?

 

[jtbell]

Here's another analogy which might be helpul. Put two people on the Earth's equator, some distance apart. For example, imagine one is at the intersection of the equator and the prime (0-degree) meridian, and the other is at the intersection of the equator and the 10-degree meridian (either east or west, take your pick). Have them both fly northward along a constant heading. They start out flying parallel to each other, but gradually their paths start to converge, even though neither of them turns left or right. When they reach the North Pole, bang! They run into each other! The curvature of the Earth's surface "sucked" them together.

 

[MeJennifer]

Here's another analogy which might be helpul. Put two people on the Earth's equator, some distance apart. For example, imagine one is at the intersection of the equator and the prime (0-degree) meridian, and the other is at the intersection of the equator and the 10-degree meridian (either east or west, take your pick). Have them both fly northward along a constant heading. They start out flying parallel to each other, but gradually their paths start to converge, even though neither of them turns left or right. When they reach the North Pole, bang! They run into each other! The curvature of the Earth's surface "sucked" them together.

Sorry but the fact that test masses on parallel lines converge in a gravitational field is no explanation at all for why they get sucked into the center. The trajectories converge because of the curvature of space-time but that by itself is no explanation for why they get drawn in.

 

[Line]

Here's the thing, with no momentum no object should move no matter how curved the space.

 

[jtbell]

The trajectories converge because of the curvature of space-time but that by itself is no explanation for why they get drawn in.

Why do you not consider spacetime curvature to be an explanation? What sort of thing would you accept as an explanation?

 

[MeJennifer]

Why do you not consider spacetime curvature to be an explanation?

Just because something is curved does not explain that something on it got to move.

What sort of thing would you accept as an explanation?

Think about it, consider two objects, a planet and a test particle far removed from the planet. The planet and the test particle are approaching each other, initially the approaching speed is very slow but it gets increasingly faster. Now both objects will be in an inertial motion, so we can rule out some kind of force, and both objects will experience that the other object approaches in an accelerating fashion. Now how could we possibly explain that by some sort of curvature? Sure there is curvature, since parallel lines converge, but curvature by itself doesn't make things move.

Stetching or moving of space would be a way to explain this. A bit like the cosmic expansion.

 

[kvantti]

But curvature by itself doesn't make things move.

Actually, nothing moves in spacetime; there exists only worldlines. The analogy presented by jtbell is very useful because it is 100% analogous to spacetime. The paths of the aeroplanes can be thought to describe geodesics and because the geometry (the surface of the planet) is curved in a particular way, the worldlines of the planes collide. It is the same in the case of spacetime: mass/energy curves spacetime in a such way that the worldlines of two objects eventually collide; even tho the worldlines of the objects follow the straightest paths in spacetime.

Motion is an illusion of spacetime. Accelerating motion induced by "gravity" is an illusion of curved spacetime.

 

[MeJennifer]

The analogy presented by jtbell is very useful because it is 100% analogous to spacetime. The paths of the aeroplanes can be thought to describe geodesics and because the geometry (the surface of the planet) is curved in a particular way, the worldlines of the planes collide.

Looks to me that it is simply an analogy for geodesic deviation, not the "sucking" effect of gravity.

 

[pervect]

Why do you think there is a difference between geodesic deviation and the "sucking" effect of gravity?

 

[MeJennifer]

Why do you think there is a difference between geodesic deviation and the "sucking" effect of gravity?

It seems to me that geodesic deviation is not the complete picture.

Feel free to explain how something could change position by only using curvature.

Consider two points with a given distance on a manifold, now make any kind of curvature, would that cause them to come together or merge?

 

[pervect]

Imagine a flat space-time diagram. Geodesics on the diagram are straight lines. If we have two obsevers starting at the same point, both following geodesics, they separate and never meet again.

Now, imagine a space-time diagram drawn on a sphere. Let the lattitude on the sphere represent the time dimension. Then two objects with different velocityes would follow different geodesics on the sphere. Let both objects start at the south pole. Then their geodesics separate for a while, reach a maximum distance, and then rejoin at the north pole. Geodesic deviation of space-time has caused what appears to be a force which causes the observers to accelerate towards each other.

This is a standard texbook example.

 

[kvantti]

Yay.

It is important to notice that it isn't only the 3D space we experience that curves, but the whole 4D spacetime. Curvature of the time dimension is as important as curvature of the three spatial dimensions.

 

[MeJennifer]

Now, imagine a space-time diagram drawn on a sphere. Let the lattitude on the sphere represent the time dimension. Then two objects with different velocityes would follow different geodesics on the sphere. Let both objects start at the south pole. Then their geodesics separate for a while, reach a maximum distance, and then rejoin at the north pole. Geodesic deviation of space-time has caused what appears to be a force which causes the observers to accelerate towards each other.

This indeed describes geodesic deviation.
But where is the center of gravity in this model? You only describe how they approach each other in time, you do not describe how they approach the center of gravity.

 

[MeJennifer]

It is important to notice that it isn't only the 3D space we experience that curves, but the whole 4D spacetime.

That is correct, 3D space is curved as well, unfortunately some people treat it as if it is flat and then "explain" GR effects in Newtonian terminology such as "changing" of frequencies, "deflection" of light or "accelerating" motions of free falling objects near a gravitational field.

Curvature of the time dimension is as important as curvature of the three spatial dimensions.

True, but isn't is odd that in case of the expansion of the universe time is treated as if it is flat like in the Robertson-Walker Metric?

 

[kvantti]

True, but isn't is odd that in case of the expansion of the universe time is treated as if it is flat like in the Robertson-Walker Metric?

The Robertson-Walker metric is an approximation, which is accurate enough to describe the time-evolution of the whole universe.

 

[freewanderer]

Gravity is what bends spacetime. Larger bodies have a larger force of gravity. The curvature of spacetime explains why objects having a gravitational pull behave the way they do next to others. The Earth is in the dip of the sun. The sun makes a dip in spacetime because of gravity. What is gravity and why does it pull, suck, and in general gravitate? This is where quantum theories are going. There is talk of gravitons and strings. Tiny vibrating energy strings create a force that when multiplied with other tiny vibrating strings creates the very obvious force of gravity. Also, the acceleration and motion of objects may result from unequal forces of gravity, big sun, small earth, or from other forces and energies.

 

[Bussani]

Gravity is what bends spacetime. Larger bodies have a larger force of gravity. The curvature of spacetime explains why objects having a gravitational pull behave the way they do next to others. The Earth is in the dip of the sun. The sun makes a dip in spacetime because of gravity.

Isn't it the other way around in General Relativity? Mass-energy bends spacetime and gravity is a result of spacetime being bent.

 

[talk2glenn]

But where is the center of gravity in this model?

The center of mass.

Then their geodesics separate for a while, reach a maximum distance, and then rejoin at the north pole. geodesic deviation of space-time has caused what appears to be a force which causes the observers to accelerate towards each other.

If I understand Jennifer's complaint correctly, it has to do with initial velocity. They agree that curved space is sufficient explanation for why an object with an initial velocity would tend to "fall towards" a massive body as it follows an initial geodesic.

Their concern seems to be that if you placed an object into space but at rest (no initial velocity), a curvature absent a motivating force is insufficient to explain the resultant acceleration of the object towards nearby masses.

The answer seems to be that all objects want to follow their natural geodesics in spacetime, by nature. The Earth isn't acting to "suck" objects in, but rather to push them out, by acting as an external force against the inertial velocity towards the "center" of spacetime (that is, right now I am traveling along a geodesic path in spacetime, but the floor - the Earth - is pushing back, keeping my spatial position static). In the Earth's absence, we are still trying to "fall" towards the "center" of 4D spacetime. Unfortunately, it is a highly abstract explanation; I can understand why its difficult to accept.

It may help to stop thinking of a conventionally non-moving object as at rest in spacetime. Even if the object has no apparent velocity in 3D space (its not going up or down, left or right) we would agree that the object has momentum through time. So in 4D space, nothing is ever at rest (just as nothing stops moving through time from our primitive, 3D perspective); your objects are always following a vector line, which includes a time component as well as a spatial component.