How does spactime curvature cause acceleration?

[KingMerv00]

I have a basic understanding of General Relativity:

The example that is always highlighted is the "bowling ball on the trampoline". When you roll a marble on the trampoline, its path is altered by the bends in the material. That's all well and good, but I don't see how this model applies to real life.

Let's say a moon is kept stationary relative to the Earth by a giant hand. If the hand lets go, the moon and the earth pick up speed and collide with each other. Why? In the trampoline model, the marble accelerates towards the center of the earth because of the effects of gravity. What does the moon accelerate towards? A higher demension?

What am I missing?

[goodgirlonhere]

Remember Apollo 13? They used the moon's gravity as a "slingshot" to bring them back to earth. I believe what your describing is alot like what they did. I am not sure of all the theories involved but I do know that is a good example of what happens.

[KingMerv00]

Remember Apollo 13? They used the moon's gravity as a "slingshot" to bring them back to earth. I believe what your describing is alot like what they did. I am not sure of all the theories involved but I do know that is a good example of what happens.

I understand how spacetime curvature can alter the path of an object that is already in relative motion; it merely follows the shape of space.

What I don't get is how two objects that are stationary relative to each other are forced to move together.

[uruk]

Gravity can also be describe as a force as well as geometry. Gravity is always attractive. Two objects that are stationary to each other will still attract each other because of the force of gravity. Anything with mass has a gravitational field that is always attracting everything else. Anything near it will be attracted to it anything passing by will alter it's path accordingly. By how much depends on mass.

[KingMerv00]

Gravity can also be describe as a force as well as geometry. Gravity is always attractive. Two objects that are stationary to each other will still attract each other because of the force of gravity. Anything with mass has a gravitational field that is always attracting everything else. Anything near it will be attracted to it anything passing by will alter it's path accordingly. By how much depends on mass.

My (poor) understanding of GR is that the "force" of gravity is just a name we give the the acceleration caused by the curvature of space. Am I wrong?

[SpaceFluffer]

No, you're correct. General Relativity reveals the force we call gravity to be a local consequence of a global curvature of spacetime.

Unfortunately the 'bowling ball on the trampoline' analogy is misleading at best, since it depicts the space-time curvature as a 2-d spacial curvature. I feel it's best to understand this stuff in terms of an object's worldline (http://en.wikipedia.org/wiki/World_line).

All objects follow timelike geodesics through spacetime, and the presence of a massive object causes the spacetime to become curved such that the path of the object (it's worldline) becomes changed accordingly. Locally this change in path is experienced as a force.

A stationary object still has a timelike worldline (i.e. it still moves through time!), and the presence of a mass still affects that worldline. I've seen some authors refer to this as 'tipping the light cone' (http://en.wikipedia.org/wiki/Light_cone). It's almost like the object is still going 'straight ahead', but the direction that 'straight ahead' takes you has changed.

[Soapy Sam]

Back to basics.
For spacetime to "curve", it must be embedded in something not curved, otherwise the term "curve"would seem to be meaningless.

The "rubber sheet " model confuses me too. You know- "the cannonball distorts space , so the orbiting ball bearing runs downhill into the well."
This may be good science, but it's lousy metaphor. It uses gravity to explain gravity. I know the ball bearing runs downhill. The question is, why?

[Melendwyr]

Back to basics.
For spacetime to "curve", it must be embedded in something not curved, otherwise the term "curve"would seem to be meaningless. Not mathematically true. Curvature is a property of the space, not of the space relative to anything else.

The "rubber sheet " model confuses me too. You know- "the cannonball distorts space , so the orbiting ball bearing runs downhill into the well."
This may be good science, but it's lousy metaphor. It uses gravity to explain gravity. I know the ball bearing runs downhill. The question is, why? SpaceFluffer gave a good explanation of this above.

The distortion of gravity takes place not only in three dimensions, but in the fourth as well. This is what's responsible for the "slowing" of time experienced inside a gravity well.

Think of it this way: if the object isn't moving, then its position will remain the same as time goes forward. If space AND time is bent by a mass, the object will remain in the same position, but there "whereness" of that position will be twisted towards the mass.

[Schneibster]

OK, here's the deal.

In 3-D space, there are three planes of rotation: x-y, x-z, and y-z. If you add one more dimension, to make 4-D spacetime, you add three more planes: x-t, y-t, and z-t. So how do we interpret rotations in those three additional planes?

Simple. Look at the Lorentz transform. Note that a velocity causes the interpretation of your x as partly their x and partly their t; the obvious answer is therefore that we interpret a rotation in one or more of these three additional planes as a velocity. That's how the Lorentz transform transforms space into time and vice-versa.

Now, if we add a curve in spacetime due to the force of gravity, that curve will cause a rotation of any object that is within the field, according to any observer outside the field, right? That's what we mean when we say, "gravity curves spacetime." And that rotation can't be in any of the three "ordinary" x-y, x-z, or y-z planes, since that would mean that space was curved by gravity, but time wasn't, and we've already said that gravity is spacetime curvature. So that means that the rotation must be in x-t, y-t, or z-t, or some combination of the above- but we also said that rotation in those planes is velocity; and, of course, what we are really talking about is the continuous addition of rotational angle, in other words, continuous variation of velocity, which we of course define as acceleration.

And that's how a gravity field results in an acceleration. See how simple it is?

[epepke]

Back to basics.
For spacetime to "curve", it must be embedded in something not curved, otherwise the term "curve"would seem to be meaningless.

The "rubber sheet " model confuses me too. You know- "the cannonball distorts space , so the orbiting ball bearing runs downhill into the well."
This may be good science, but it's lousy metaphor. It uses gravity to explain gravity. I know the ball bearing runs downhill. The question is, why?

As Melendwyr pointed out, this is not mathematically supportable. Curvature can be represented as a completely intrinsic concept. All you have to do is try to construct a rectangle or triangle or something using the Euclidean laws. If you find that the length of the sides and the angles don't match up according to Euclid, you can conclude that the space is curved.

You can try to make a rectangle in spacetime under the influence of gravity. It's easy. Put a clock way up high, maybe on the top of the Sears Tower or something, and put a clock at the bottom. One dimension is the vertical displacement of the two clocks. That's the left and right sides. The top side is the elapsed time of the top clock. The bottom side is the elapsed time of the bottom clock. Then, say, you're trying to make a rectangle with a vertical length of 1450 feet and a time length of 10 seconds. However, the top clock runs faster than the bottom clock, due to gravity. So the sides of the rectangle won't quite meet. From that alone, we can conclude that spacetime is curved.

As for the OP, well, why is something that nobody knows. We know that it does, and that it does so based on the flux of energy/momentum. But we don't know why. Perhaps m-brane theory will provide an answer.

[CapelDodger]

Simple. Look at the Lorentz transform.
:clap: I'm gonna steal that.

[Soapy Sam]

As Melendwyr pointed out, this is not mathematically supportable. Curvature can be represented as a completely intrinsic concept. All you have to do is try to construct a rectangle or triangle or something using the Euclidean laws. If you find that the length of the sides and the angles don't match up according to Euclid, you can conclude that the space is curved.- Epepke

I can't get my head round this. Curvature is just a measure of departure from a chosen baseline. It's not an intrinsic property of anything. Without a baseline, how could it even be measured?
Example- On a sphere, three lines from pole to equator, a quarter around the equator and back to the pole, create a triangle of 270 degrees. Euclidean rules don't apply , because they define a triangle as having three straight sides, which this does not. The sides are curves drawn by the intersection of three flat planes with a spherical surface. But the spherical surface itself is defined relative to the three orthogonal planes. It's only spherical if you can get back and see it from a distance in space. Close up, it's flat. The old Columbus conundrum.


And that's how a gravity field results in an acceleration. See how simple it is?-Schneibster.

No, though I maybe had a brief glimmer. I lack two important tools- an ability to visualise and a mathematical understanding of anything at all. This is why I get hung up on definitions of words. I 'm struggling to understand how curvature can be an intrinsic property of anything as opposed to a relative property measured against a non curved baseline.

One step at a time, if we may.
If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?

If I don't get back to this till Saturday, it's not lack of interest. I'll be transiting 3-space for a day or two.

[luvhumility]

I like Schneibster's explanation, this is very good!

Now, if we add a curve in spacetime due to the force of gravity, that curve will cause a rotation of any object that is within the field, according to any observer outside the field, right? That's what we mean when we say, "gravity curves spacetime." And that rotation can't be in any of the three "ordinary" x-y, x-z, or y-z planes, since that would mean that space was curved by gravity, but time wasn't, and we've already said that gravity is spacetime curvature. So that means that the rotation must be in x-t, y-t, or z-t, or some combination of the above- but we also said that rotation in those planes is velocity; and, of course, what we are really talking about is the continuous addition of rotational angle, in other words, continuous variation of velocity, which we of course define as acceleration.
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I think this may be called conservation of angular momentum? I am not sure. If one thinks of it... this spin and rotational energy storage happens from the macro (cosmos), all they way down to the (micro) smallest atoms and particles. I look at it like all things are rotating around all other things as well as spinning TOO at all levels of the system big to small. They store more energy and gain more "mass" or "energy" as they spin and rotate faster and "compress". Maybe Its like energy in a 4 dimensional system with the spins and rotations coserving energy and being the 4th Dim or similar. Then as the system as a whole loose's energy they all slow their spins and their orbits move apart. Anyspin vs a vs or Someting similar to that???

lh

[luvhumility]

One step at a time, if we may.
If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?
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I think the answer to this is no, i think its less than 180...? not sure.

[Melendwyr]

No, though I maybe had a brief glimmer. I lack two important tools- an ability to visualise and a mathematical understanding of anything at all. This is why I get hung up on definitions of words. I 'm struggling to understand how curvature can be an intrinsic property of anything as opposed to a relative property measured against a non curved baseline. If the interior angles of a triangle add up to 180 degrees, it's in Euclidean space. If they don't, it isn't.

This is true if one of the basic axioms of Euclidean geometry (given a line on a surface and a point not on that line, there is one and only one line that can be drawn through that point that is parallel to the other line) is violated. There doesn't have to be an "absolute space" that the surface (or volume, or whatever) is within. It's just a property of how the points can be connected.

[luvhumility]

One step at a time, if we may.
If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?
-------------------------------
I think the answer to this is no, i think its less than 180...? not exactly sure.

I also think your right that the normal euclid math wont do the trick here.

[luvhumility]

Yuk! I bet this is where the 3D+ heavy duty calculus and vector math must be used…?

[Schneibster]

No, it's still a Euclidean space- you have to deform it in two dimensions for the angles not to sum to 180 degrees. And you can't do that without tearing the paper. ;)

[Schneibster]

I like Schneibster's explanation, this is very good!Thanks!
I think this may be called conservation of angular momentum? I am not sure. If one thinks of it... this spin and rotational energy storage happens from the macro (cosmos), all they way down to the (micro) smallest atoms and particles. I look at it like all things are rotating around all other things as well as spinning TOO at all levels of the system big to small. They store more energy and gain more "mass" or "energy" as they spin and rotate faster and "compress". Maybe Its like energy in a 4 dimensional system with the spins and rotations coserving energy and being the 4th Dim or similar. Then as the system as a whole loose's energy they all slow their spins and their orbits move apart. Anyspin vs a vs or Someting similar to that???

lhBe careful- velocity in threespace-plus-time is a rotational static angle in fourspacetime, not a continuous rotation, like a spin. A revolution about a plane that is defined to include the t-axis in its plane is an acceleration, not a velocity.

Not only that, but rotation in a hyperbolically symmetric spacetime doesn't behave like a rotation in a spherically symmetric space; a "right angle" in a hyperbolic spacetime is an infinite number of degrees away from the base angle; which is why it takes an infinite amount of energy to go the speed of light. Think about it: if you rotate all your "motion in time" into "motion in space," then you will be going the speed of light, and an observer will see you experiencing no time. This requires infinite energy, and is in fact impossible. The rotation to go the "opposite direction in time" is an undefined operation in hyperbolic fourspacetime.

[Melendwyr]

You already understand how a distorted geometry can alter the path of a moving object, right, Soapy?

Well, a "stationary" object is still moving forward in time. Just as the distorted geometry causes movement through the three dimensions of space to warp, it causes movement in the time dimension to bend, too. As a straight course in space is bent, so is a straight course through time. Imagine looking at a curved line one point at a time - you see a single point that seems to move. That's what happens when gravity attracts something - you're seeing a 3D slice through a 4D curved line, which appears to be a moving point.

[CapelDodger]

Thanks!
Be careful- velocity in threespace-plus-time is a rotational static angle in fourspacetime, not a continuous rotation, like a spin. A revolution about a plane that is defined to include the t-axis in its plane is an acceleration, not a velocity.

Not only that, but rotation in a hyperbolically symmetric spacetime doesn't behave like a rotation in a spherically symmetric space; a "right angle" in a hyperbolic spacetime is an infinite number of degrees away from the base angle; which is why it takes an infinite amount of energy to go the speed of light. Think about it: if you rotate all your "motion in time" into "motion in space," then you will be going the speed of light, and an observer will see you experiencing no time. This requires infinite energy, and is in fact impossible. The rotation to go the "opposite direction in time" is an undefined operation in hyperbolic fourspacetime.

If you've got it, flaunt it. If I ever go into Space, willya come with? I'd feel a lot more relaxed.

[Schneibster]

Melendwyr, nicely done. Yes, that's another way of looking at the same thing.

[Melendwyr]

Thanks, Schneibster.

Now if I could only grasp how Relativity makes it possible to derive magnetism from rotating electric field sources... :c(

[epepke]

No, it's still a Euclidean space- you have to deform it in two dimensions for the angles not to sum to 180 degrees. And you can't do that without tearing the paper. ;)

Or having it be a sheet of rubber.

Edited to add: you can make a quilt with negative curvature. Base it on equilateral triangles, but have seven rather than six triangles come to a point at every point.

[Dr Adequate]

Let's say a moon is kept stationary relative to the Earth by a giant hand. If the hand lets go, the moon and the earth pick up speed and collide with each other. Why? In the trampoline model, the marble accelerates towards the center of the earth because of the effects of gravity. What does the moon accelerate towards? A higher demension? Well, what would happen if you put two bowling balls on a trampoline?

They'd roll together, right?

The metaphor still works.

[Soapy Sam]

You already understand how a distorted geometry can alter the path of a moving object, right, Soapy?
.
Well, no. Not really. I see how a course can follow a curve in 3 space, but I don't see why (or indeed how) the space itself can curve.
A ship moving in a circular arc from a to b could have taken a straight line from a to b. But it didn't. That does not mean the water curves. The course curves.

If space can twist, then how can we say that a course following that twist is curved? Curved relative to what? Any measuring instrument in that space is also curved. The measurement will come out as a straight line using any tools inside the space. It's only from outside the space that it appears curved, because we have a "pre curve" standard of straightness.

Suppose we can twist space. What happens if we twist it through 90 degrees so that left becomes up? Does a moving object get smeared somehow? Is the space now 2 dimensional?

Multi- dimensional hyperspaces are increasingly popular as a sort of multivariable analysis tool. But is it just that- a conceptual visualisation aid?
Metaphor is useful, but it's not the thing to be explained. Same goes for mathematics. And language. And common sense.

We have anything left?

Ah, jet lag and shift change in one day. Too tired to stay awake, to awake to sleep.
Your efforts are appreciated gents. One reason I never believed in free will is because I'm aware of my limits.

[69dodge]

If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?Yes, they do. A cylinder has zero intrinsic curvature.

If space can twist, then how can we say that a course following that twist is curved? Curved relative to what? Any measuring instrument in that space is also curved. The measurement will come out as a straight line using any tools inside the space. It's only from outside the space that it appears curved, because we have a "pre curve" standard of straightness.Intrinsic curvature of a space isn't measured with respect to an external "standard of straightness." It's measured by investigating the geometry of the space itself.

Suppose you construct a large triangle---a real physical triangle, made out of 2x4's or something---making sure that its sides are straight, and then you measure its angles and they don't sum to 180 degrees. How would you describe this situation? Mathematicians would describe it by saying that our 3D space is curved, by analogy with "really curved" spaces like the 2D surface of a sphere, where the same thing happens. What word we use to describe it isn't especially important, of course. The important point is that this situation is not a logical impossibility. One cannot prove mathematically that the angles of a physical triangle will sum to 180 degrees. If they don't, then they don't, and that's that.

And, according to general relativity, sometimes they don't.

[Soapy Sam]

Yes, they do. A cylinder has zero intrinsic curvature.

Intrinsic curvature of a space isn't measured with respect to an external "standard of straightness." It's measured by investigating the geometry of the space itself.

Would you agree that this is not true of the common usage of the word "curvature" ? In the real world we do not calculate curvature by equation; we measure it relative to a limiting case- the straight line, the curve whose curvature is zero. (Actually, I never yet had a shelf collapse, so my methods seem to work. At least for shelves.)

Suppose you construct a large triangle---a real physical triangle, made out of 2x4's or something---making sure that its sides are straight, and then you measure its angles and they don't sum to 180 degrees. How would you describe this situation?

Well, I'd describe it as a classic DIY bodge job. (ie I'd put it down to poor construction before blaming it on spacetime distortion). However I may use this explanation if one of my shelves ever falls down.


Mathematicians would describe it by saying that our 3D space is curved, by analogy with "really curved" spaces like the 2D surface of a sphere, where the same thing happens. What word we use to describe it isn't especially important, of course.

There we differ. If mathematicians are talking to each other, they may use words or other symbols to mean whatever they define them to mean, but
if they are talking to a non mathematician like me, I hope they will try to use standard English where possible and make it clear when they do not.

So the word "curve" here does not mean what I expect it to mean, but is in some way analogous to the common meaning of the word? I was unaware of this. How far can the analogy be stretched and in how many directions, before it ceases to be a useful analogy and becomes an example of the type of confusion it is meant to avoid?


The important point is that this situation is not a logical impossibility. One cannot prove mathematically that the angles of a physical triangle will sum to 180 degrees. If they don't, then they don't, and that's that.

Sorry. You lost me there. What situation do you mean?

And, according to general relativity, sometimes they don't.

[Art Vandelay]

I understand how spacetime curvature can alter the path of an object that is already in relative motion; it merely follows the shape of space.

What I don't get is how two objects that are stationary relative to each other are forced to move together.Ah, but they are merely stationary in three dimensional space. Every object is moving in four dimensional space, because it’s moving through time.

My (poor) understanding of GR is that the "force" of gravity is just a name we give the the acceleration caused by the curvature of space. Am I wrong?
Well, in GR, the “acceleration” caused by the curvature of space is also an illusion; in reality, objects are traveling in straight lines.

If the interior angles of a triangle add up to 180 degrees, it's in Euclidean space. If they don't, it isn't.That’s not quite true. If the angles at up to 180 degrees, that’s not enough to conclude that the space is Euclidean. You have to have it be true for every triangle, not just one.


Now, if we add a curve in spacetime due to the force of gravity, that curve will cause a rotation of any object that is within the field, according to any observer outside the field, right? That's what we mean when we say, "gravity curves spacetime." I don’t think you’re understanding it correctly. First of all, gravity doesn’t curve spacetime, gravity is the curvature of space time. Also, you’re confusing “curvature”, “curve”, and "rotation". You also seem to be thinking, like Soapy Sam, of curvature as being an extrinsic property that “really” exists in some higher dimensional space.

And that rotation can't be in any of the three "ordinary" x-y, x-z, or y-z planes, since that would mean that space was curved by gravity, but time wasn't, and we've already said that gravity is spacetime curvature.That doesn’t make any sense. Just because you have a rotation in four dimensional space doesn’t mean that each dimension is involved. If I rotate something in the x-y plane, is that not a spatial rotation, because space has three dimensional and I involved only two? According to your logic, why is a rotation in the x-t plane allowed? That would be ignoring y and z, which are just as much a part of spactime as t is. Three dimensional space is a subset of spacetime, so any rotation that happens in the former happens in the latter.

Thanks!
Think about it: if you rotate all your "motion in time" into "motion in space," then you will be going the speed of light, and an observer will see you experiencing no time. But light travels though an equal amount of space and time. If you rotate all your motion in time into motion in space, then you should have no motion through time left; that is, you should be moving from point to point instantaneously, not at the speed of light.

[Art Vandelay]

Back to basics.
For spacetime to "curve", it must be embedded in something not curved, otherwise the term "curve"would seem to be meaningless.You can embed it in something not curved if it helps you conceptualize it, but it's not necessary from a purely mathematical point of view.

The "rubber sheet " model confuses me too. You know- "the cannonball distorts space , so the orbiting ball bearing runs downhill into the well."
This may be good science, but it's lousy metaphor. It uses gravity to explain gravity. I know the ball bearing runs downhill. The question is, why?I had the same objection. I think that the analogy has been distorted as it's been repeated. I googled "rubber sheet relativity", and the first link had this to say:
The large ball will cause a deformation in the sheet's surface. A baseball dropped onto the sheet will roll toward the bowling ball.

The second said thisAs an analogy to Einstein's view of gravity, consider a rubber sheet which is held taut on a frame. It is flat, so if you roll a marble across it, the marble will follow a straight line. Now drop a heavy ball bearing onto the rubber sheet. The ball bearing is massive enough to cause a depression in the rubber sheet around it. Now roll a marble across the rubber sheet, close to the ball bearing. The curvature of the rubber sheet will cause the marble to follow a curved path. A distant observer, unable to see the curvature in the rubber sheet, would say ``The ball bearing must be exerting a force on the marble, deflecting it from its straight path.''

The two may seem, at first, to be very similar, they're very different. I believe that the second one is the original, and the first a misunderstanding of the second. In the second, the ball is not curving because of gravity; it’s curving because the surface forces it to do so.

To explain what the second one is getting at, I’m going to have to include some differential geometry, so bear with me. Imagine that you have a two dimensional surface embedded in three dimensional space, and on the surface is a one dimensional curve. Now imagine that you’re traveling along that curve. Let v be your velocity vector; that is, it’s an arrow whose direction is determined by which way you’re going, and whose size is determined by how fast you’re going (that is, |v|). Now, we can choose to keep our speed constant, so the velocity vector will always be of constant length. Define a to be your acceleration vector; that is, an arrow that points in the direction that v is changing. a is actually going to be perpendicular to v, because |v| is constant (that is, since |v| is constant, then there's no acceleration is direction of travel). So one dimension is out, which means that there are two left. If we call n the vector pointing directly away from the surface, and c the vector perpendicular to both v and n, then a can be any combination of n and c. Now, anyone constrained to this surface won’t be really be able to “see” n, because it points outside the surface, a direction that doesn’t exist for this being. c, on the other hand, is within the surface, so this being will be able to see it just fine. So if we ask this being whether the curve is straight, he’s just going to look at c. This leads to a new concept: a curve is straight, as far as a particular surface is concerned, if c is zero; that is, it is allowed to curve, but only to follow the surface.

Now, getting back to the rubber sheet, the ball is going to follow a straight line within the surface. An observer looking at its motions in three dimensions is going to see it as curving, but a being looking at it from within the surface is not going to see it curve. Note that if the sheet is curved, the ball must curve to match it, but this is the only curving that it’s doing. Remember that force is mass times acceleration, so if there’s no acceleration, there’s no force. From a three dimensional point of view, we see acceleration, so we also see a force (the surface is exerting a force on the ball; otherwise the ball would just pass right trough it). But looking at it from within the surface, there is no acceleration, so the ball experiences no force. But if we introduce gravity, and say that the ball rolls down the slope because of it, then the ball will experience a force, so the first link is misstating the situation. So, to clarify: the first quote basically says that the ball is curving because of gravity. The second one, however, does not attribute the curvature to gravity, but to the curvature of the sheet.

I think a better example would be if two people both start out at the South Pole and walk in opposite directions. Eventually, they will meet again at the North Pole, even though they both walked in straight lines. They might conclude that there was some mysterious “force” that pulled them together.

I can't get my head round this. Curvature is just a measure of departure from a chosen baseline. It's not an intrinsic property of anything. Without a baseline, how could it even be measured?One thing to keep in mind is that simply because something is measured in terms of a baseline, that does not mean that it is not intrinsic. If the choice of baseline has no effect on the quantity measured, then it's still intrinsic, even if it's defined in terms of extrinsic quantities. For instance, the distance between two points is sqrt(x^2+y^2+z^2), right? Even though, in order to define x, y, and z, you have to choose a particular coordinate system, it turns out that you get the same answer no matter what.

The sides are curves drawn by the intersection of three flat planes with a spherical surface. But the spherical surface itself is defined relative to the three orthogonal planes.Can you think of a different embedding of the sphere that results in different curvature? In fact, a straight line can be defined in terms of the metric; for instance, on a sphere, one can measure all possible paths from Los Angeles to New York. The "straight line", within the spherical geometry, would be the one that contains the shortest path. There is no need to imagine involve three dimensions to define a straight line.

It's only spherical if you can get back and see it from a distance in space. Close up, it's flat. The old Columbus conundrum.Close up, it's still a sphere, it's just that one cannot tell that it's curved. Just because you don't realize it's curved doesn't mean it isn't. There's actually a more general formula for triangles: the sum of the angles is equal to pi + integral of curvature. On a sphere, the curvature is constant, so it's just pi + (curvature*area). The bigger the triangle, the more the sum of the angles will differ from pi. For instance, in your example, each angle is a right angle, or pi/2. The total angle is then 3pi/2, or pi + pi/2. So curvature*area=pi/2. Note that this triangle is 1/8 of a total sphere, and the area of the total sphere is 4(pi)r^2. So the area of this triangle is (pi/2)r^2, which means the curvature is 1/r^2.

If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?No. In a 2 dimensional manifold, you can find the curvature by drawing two orthogonal vectors tangent to the surface, finding the curvature vectors of the two tangent vectors, then taking the dot product of those two vectors. For a cylinder, there's horizontal curvature, but no vertical curvature, so the product is zero. Note that even though this defintion depends on an outside coordinate system, it comes out to be the same no matter what you pick.

Also, any surface that you can make by bending a piece of paper without stretching it by definition has zero curvature.

[Art Vandelay]

Would you agree that this is not true of the common usage of the word "curvature" ? In the real world we do not calculate curvature by equation; we measure it relative to a limiting case- the straight line, the curve whose curvature is zero. (Actually, I never yet had a shelf collapse, so my methods seem to work. At least for shelves.)Keep in mind that there is a difference between curvature of a curve and curvature of a surface.

Now, when you compare a curve to a straight line, you aren't comparing it to an external standard; a straight line, after all, is part of spacetime, and what constitutes a "straight line" is itself an intrinsic property of space, not an external standard. There's no way we can step out of our universe and say "okay, this is a straight line, let's use that as our standard".

There we differ. If mathematicians are talking to each other, they may use words or other symbols to mean whatever they define them to mean, but
if they are talking to a non mathematician like me, I hope they will try to use standard English where possible and make it clear when they do not.Seeing as how "the curvature of spacetime" has no meaning in "standard English", I think that it's rather clear that it is the mathematical definition that applies. Mathematicians don't make up new terms and definitions to confuse non mathematicians; they do it because non mathematicians don't have words for what mathematicians want to talk about.

So the word "curve" here does not mean what I expect it to mean, but is in some way analogous to the common meaning of the word?That depends on what you expect it to mean, but pretty much every mathetical term can be considered an analogy. That's what mathematics is: the study of analogies. When mathematicians speak of volume, or space, or surface, etc., they are talking about mathematical concepts which are analogous to the common meanings of those words. "Standard English" gives words such vague, ambiguous, contradicatory, and incomplete definitions that mathematicians pretty much have to make up new definitions.

Sorry. You lost me there. What situation do you mean?The proof of this principle relies on assumptions which may not be true in the real world. It is a statement about the mathematical abstraction of a triangle, not physical triangles.

[69dodge]

If I draw a perfect, Euclidean right triangle on a flat piece of paper, then roll that sheet into a tube, then I measure the angles of the triangle. Do they sum to 180 degrees, or not?
No.I'm pretty sure you meant to say "yes," because you do go on to say, correctly, that the cylinder isn't curved.

[69dodge]

Would you agree that this is not true of the common usage of the word "curvature" ? In the real world we do not calculate curvature by equation; we measure it relative to a limiting case- the straight line, the curve whose curvature is zero. (Actually, I never yet had a shelf collapse, so my methods seem to work. At least for shelves.)Yes, I agree.

Well, I'd describe it as a classic DIY bodge job. (ie I'd put it down to poor construction before blaming it on spacetime distortion).Sure, that makes perfect sense, in practice. The difference from 180 degrees due to spacetime curvature is ridiculously small for reasonably sized shelves.

So the word "curve" here does not mean what I expect it to mean, but is in some way analogous to the common meaning of the word?Yes, that's correct.

The important point is that this situation is not a logical impossibility. One cannot prove mathematically that the angles of a physical triangle will sum to 180 degrees. If they don't, then they don't, and that's that.
Sorry. You lost me there. What situation do you mean?One in which a physical triangle exists whose angles don't add up to 180 degrees.

[Soapy Sam]

Gentlemen (or whatever). Your trouble is appreciated. I need to ponder upon this. I feel a paradigm shifting. (Ponderously).

[Art Vandelay]

I'm pretty sure you meant to say "yes," because you do go on to say, correctly, that the cylinder isn't curved.
Oops. I mean, um, I was replying "no" to the "or not" part. Yeah, that's it.