## gravity caused by curved space - am I right?

I went over this with someone here a while ago and I think they became exhausted with me. Anyway, I went away to think about it. Here is what I have so far.

I was told that gravity is caused by warped space time (according to, is it GR or SR). I was told that when you move into a curve, you accelerate. This is what was really confusing me. I tried to picture something moving faster because it was going from a straigh path to a curved one but I just wasn't seeing it. How does moving along a curve translate to increasing speed? Then it dawned on me that one thing I could see was that if the curve was getting tighter and tighter, as 'twere, one thing that does increase and that is the rate at which the object in question is changing directions! So the picture I formed in my head is this: Imagine the earth by itself in space (to avoid distractions). Now pop something into existence, say, a person, some ways away from the Earth. Now this person, having just popped into existence, instead of just floating in space will start to move towards the earth, slowly at first and then faster and faster, making a straight path. Now, this straight line being traced is what we see, but the reason the person is accelerating is because this is not the whole picture. What we see as a straight line path is actually a person changing directions because they are actually travelling in a curve and the reason they are accelerating is because the curve they are caught on is getting more and more curved. In other words, the faster speed we see is the increased rate the person is changing direction on the ever tightening curve.

Could someone please tell me if this is what the whole deal is about or have I descended into madness? Many many thanks.

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 Recognitions: Gold Member Science Advisor Staff Emeritus Yes, it is true that in General Relativity, gravity is pictured as a "warping" of space. In classical, Newtonian, physics, all objects move in a straight line unless they are affected by a "force". General Relativity replaces the straight line with the "geodesic" of a curved surface. The "force" of gravity is replaced by the fact that geodesics are no longer straight lines. "I was told that when you move into a curve, you accelerate. This is what was really confusing me. I tried to picture something moving faster because it was going from a straigh path to a curved one but I just wasn't seeing it." In physics, "accelerate" does not mean "move faster". "Accelerate" means "change velocity". Slowing down (in common parlance "decelerate") is also a form of acceleration. In fact, since velocity is a "vector" quantity (having both speed and direction) any time you change direction, you are changing your velocity and so are "accelerating". If you move around a circle, with [b]constant[b] speed, since you are constantly changing direction, you are accelerating. Since going around a curve necessarily means changing direction, going around a curve necessarily means "accelerating" but not "going faster".

 Originally posted by Vosh I was told that gravity is caused by warped space time (according to, is it GR or SR).
You were told wrong. Gravity is not caused by curved spacetime. Curved spacetime is just another term for tidal force and tidal force is not the cause of gravity. In fact you can have a region of space where there is a gravitational field where there are no tidal forces in that region. That's what a uniform gravitational field is and that is why Einstein refered to it in his principle of equivalence, i.e. Eintein said
 A uniform gravitational field is equivalent to a uniformly accelerating frame of reference.
You wrote
 I was told that when you move into a curve, you accelerate. This is what was really confusing me. I tried to picture something moving faster because it was going from a straigh path to a curved one but I just wasn't seeing it. How does moving along a curve translate to increasing speed?
When you're sitting in your chair in front of your computer then you're at rest in a gravitational field. Being at rest in a gravitational field is (locally) equivalent to accelerating with respect to an inertial frame of reference. Do not confuse the curved spatial path of a particle with spacetime curvature. You can have a spatially curved path in flat spacetime.

Geodesics should not be thought of as straight lines. Think of them as lines of extremal lenght. Or if you wish think of it as a "straightest possible line." But a path, even a geodesic, on a curved surface surface is not a straight line. However if one resides within such a surface then geodesics would appear as straight lines in that sense.

Keep in mind that geodesics are the "straightest possible lines" not in space but in space*time*

## gravity caused by curved space - am I right?

 Originally posted by HallsofIvy Yes, it is true that in General Relativity, gravity is pictured as a "warping" of space. In classical, Newtonian, physics, all objects move in a straight line unless they are affected by a "force". General Relativity replaces the straight line with the "geodesic" of a curved surface. The "force" of gravity is replaced by the fact that geodesics are no longer straight lines.

Not quite, you must be very careful. Objects in free fall always follow geodesics. The fact that spacetime is curved doesn't mean that the line is no longer straight. It is, in fact, a straight line on a curved surface (the same way lines of latitude and longitude on a sphere are straight lines). They "look" curved, because we like to associate the term "straight" with flat space(time) only. But it is important to remember that they are straight.

The fact that spacetime is curved and geodesics may not behave the way they do in flat spacetime is what manifests itself as a deflection from an otherwise straight path.

 Recognitions: Gold Member Staff Emeritus the same way lines of latitude and longitude on a sphere are straight lines Sartor resartus. Lines of latitude aren't geodesics on the sphere, except for one, the equator. Lines of longitude are geodesics ("great circles"); the use of the term "straight line" for them is, well,...specialized.

 Originally posted by HallsofIvy In physics, "accelerate" does not mean "move faster". "Accelerate" means "change velocity". Slowing down (in common parlance "decelerate") is also a form of acceleration.
I know. I was told that a curved path causes acceleration. So I though, ok, a change in velocity, it's either slowing down or speeding up. I don't see how moving in a curved path makes you speed up or slow down, so I don't see how following a curving path in spacetime causes you to accelerate towards the earth.

 In fact, since velocity is a "vector" quantity (having both speed and direction) any time you change direction, you are changing your velocity and so are "accelerating".
Right, and when I read that I thought, ok, the direction is changing, but the speed isn't. Same speed, different direction which now turns out, according to my research, to be the definition of velocity. What this sounds like is that I can move at a uniform speed, change direction and announce, "look, I'm accelerating". Kinda confusing.

 [b] If you move around a circle, with [b]constant speed, since you are constantly changing direction, you are accelerating. Since going around a curve necessarily means changing direction, going around a curve necessarily means "accelerating" but not "going faster".
So I'm back to, why do you speed up as you approach the earth?

 Originally posted by pmb_phy You were told wrong. Gravity is not caused by curved spacetime. Curved spacetime is just another term for tidal force and tidal force is not the cause of gravity. In fact you can have a region of space where there is a gravitational field where there are no tidal forces in that region. That's what a uniform gravitational field is and that is why Einstein refered to it in his principle of equivalence, i.e. Eintein said You wrote When you're sitting in your chair in front of your computer then you're at rest in a gravitational field. Being at rest in a gravitational field is (locally) equivalent to accelerating with respect to an inertial frame of reference. Do not confuse the curved spatial path of a particle with spacetime curvature. You can have a spatially curved path in flat spacetime. Geodesics should not be thought of as straight lines. Think of them as lines of extremal lenght. Or if you wish think of it as a "straightest possible line." But a path, even a geodesic, on a curved surface surface is not a straight line. However if one resides within such a surface then geodesics would appear as straight lines in that sense. Keep in mind that geodesics are the "straightest possible lines" not in space but in space*time*
Whoa. Tidal forces? First time I've heard that expression! I know that accelerating with respect to an inertial frame is equivalent to standing on the earth. I know that a geodesic is the shortest path between two points on a dome shaped surface. What I don't know is why I move faster and faster as I fall towards the Earth. Someone on another thread suggested, I thought, that it's because I'm actually tracing a curved path in spacetime and that because I'm residing in that curved space time it appears to be a straight line to me and since I'm on this spacetime geodesic, I'm accelerating, which I took to mean getting faster and faster as I approach landing only now I find out that the definition of acceleration is merely changing direction. So while I'm changing direction on this curve I'm not actually accelerating (as oppose to decelerating) and so I'm left with my question which is why is my speed getting faster and faster.

Hmm, what I'm sort of seeing is that residing on this geodesic means I'm changing direction and I take it this is why I started moving towards the earth in the first place. But you say that this isn't so and mentioned something called "tidal force"...

I'll leave it there.

 Originally posted by pmb_phy You were told wrong. Gravity is not caused by curved spacetime. Curved spacetime is just another term for tidal force and tidal force is not the cause of gravity. In fact you can have a region of space where there is a gravitational field where there are no tidal forces in that region.
ummm...what are you talking about tidal forces? Mass warps spacetime...this is gravity...not sure what you mean in your post...in fact i ahve no clue where tidal forces came from. PLease explain.

 Originally posted by franznietzsche ummm...what are you talking about tidal forces? Mass warps spacetime...this is gravity...not sure what you mean in your post...in fact i ahve no clue where tidal forces came from. PLease explain.
Tidal forces are just another way to say "spacetime curvature" = they mean the exact same thing. Spacetime curvature refers to the geodesic deviation, i.e. two geodesics which start out parallel do not remain parallel. Physically this manifests itself as two particles in free-fall accelerating relative to each other. Thus two particles will cross paths even when they start out moving parallel to each other. That's why Kip Thorne wrote
 Thus, Einstein and Newton, with their very different viewpoints on the nature of space and time, give very different names to the agent that causes the crossing. Einstein calls it spacetime curvature; Newton calls it tidal gravity. But there is just one agent acting. Therefore, spacetime curvature and tidal gravity must be precisely the same thing, expressed in different languages.
If there is mass at a point X then spacetime is curved at that point. However if there is a point Y which is different than X which is near X but for which there is no matter at Y may still have a gravitational field at Y.

Here is a Newtonian example: If there is mass at point X then the Newtonian tidal force tensor does not vanish at X. But that doesn't mean that the gravitational field in nearby points must have a tidal gradient there. Here is such an example

http://www.geocities.com/physics_wor...rav_cavity.htm

Here is a derivation of tidal accelerations in Newtonian language
http://www.geocities.com/physics_wor...rce_tensor.htm

Here is a derivation of tidal accelerations in Einsteinian language
http://www.geocities.com/physics_wor..._deviation.htm

 Originally posted by Vosh What I don't know is why I move faster and faster as I fall towards the Earth.
Nobody else knows why either. Einstein did not explain why. He just described it differently and much more accurately. But general relativity does not exlain why things accelerate in free-fall in a gravitational field. But to see all this clearly keep in mind the difference between a gravitational field and spacetime curvature. Picture yourself in an accelerating frame of referance. You will be unable to distinguish it from a uniform gravitational field. Yet there is no spacetime curvature since spacetime curvature refers to the non-uniformities in a gravitational field.

 Originally posted by pmb_phy Gravity is not caused by curved spacetime.
Actually, without Riemannian spacetime curvature one is left with the spacetime of special relativity where the affine connections can be globally transformed away. This leaves one with no gravitational forces anywere in the universe. On the other hand if one has this spacetime curvature the affine connections can not be globally transformed away and one always haves gravitational forces somewhere in the universe. This is sufficient to demonstrate a causal relation.

 Originally posted by pmb_phy Curved spacetime is just another term for tidal force ...
No, the Riemann tensor depends on both first order derivatives of the affine connections and products of the connections without derivatives. Even if one has a nonzero Riemann tensor one can always choose a frame for which the derivative terms vanish even though the product terms do not.

 Originally posted by pmb_phy Eintein said, A uniform gravitational field is equivalent to a uniformly accelerating frame of reference.
He was referring to the Newtonian field, not general relativity's field of spacetime curvature.

There has been an unfortunate trend in GR where the deflection of particles was attempted to have been explained in geometrical terms. However GR has never been more geometrical than any other theory. The analogy with curved surfaces has led people to the erroneous belief that it was the actual cause of gravity rather than a different way to describe gravity. But GR is just as geometrical as EM. As Einstein said
 I do not agree with the idea that the general theory of relativity is geometrizing physics or the gravitational field. The concepts of physics always have been geometrical concepts and I cannot see why the gik field should be called more geometrical than f.i. the electromagnetic field or the distance of two 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 is in a letter to Lincoln Barnett dated 1948.

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 Originally posted by Vosh I was told that when you move into a curve, you accelerate. This is what was really confusing me. I tried to picture something moving faster because it was going from a straigh path to a curved one but I just wasn't seeing it. How does moving along a curve translate to increasing speed?
If the world-line of a particle is curved, then acceleration exists for this particle.

Consider a 1-D space. The worldline is drawn in 2-D space-time, one D for space, and the other D for time. The slope of the world-line is the speed. So, if the world-line is curved, then its slope changes, therefore the speed changes.

For higher D space, even if the slope of spatial displacement wrt time displacement is constant, the world-line can have a change in the dx/dy slope, for instance. As you pointed out, this would be a change in direction, which also cannot happen in the absence of acceleration. Just don't forget about the space to time slope.

In other words:
The dx/dy, dy/dz, and dz/dx slopes are direction. If they change, then there is acceleration by virture of changing direction.
The dl/dt slope is speed (dl2 = dx2 + dy2 + dz2). If it changes, then there is acceleration by virtue of changing speed.

 Originally posted by turin If the world-line of a particle is curved, then acceleration exists for this particle.
If the worldline is a geodesic then the acceleration exists for the observer, not the parricle. Whether a worldline is curved or not depends on one's choice of coordinates. I.e. consider a free-particle in flat spacetime. In an inertial frame of referance the worldline in a spacetime diagram will be straight if the spatial coordinates are Cartesian. However switch to a coordinate system which corresponds to a frame of reference which is accelerating relative to the first. If the spatial coordinates are chosen to be Cartesian then the worldline will now be curved. This is equivalent to plotting a straight line in the xy-plane and plottint the same line in the r-theta (i.e. polar) coordinate system when r and theta are plotted on linear orthogonal axes.

 Originally posted by pmb_phy You were told wrong. Gravity is not caused by curved spacetime. Curved spacetime is just another term for tidal force and tidal force is not the cause of gravity. In fact you can have a region of space where there is a gravitational field where there are no tidal forces in that region. That's what a uniform gravitational field is and that is why Einstein refered to it in his principle of equivalence, i.e. Eintein said You wrote When you're sitting in your chair in front of your computer then you're at rest in a gravitational field. Being at rest in a gravitational field is (locally) equivalent to accelerating with respect to an inertial frame of reference. Do not confuse the curved spatial path of a particle with spacetime curvature. You can have a spatially curved path in flat spacetime. Geodesics should not be thought of as straight lines. Think of them as lines of extremal lenght. Or if you wish think of it as a "straightest possible line." But a path, even a geodesic, on a curved surface surface is not a straight line. However if one resides within such a surface then geodesics would appear as straight lines in that sense. Keep in mind that geodesics are the "straightest possible lines" not in space but in space*time*
Well, you responded to someone else's response using grammar and terms I couldn't follow in the least, so I'm replying again in hopes that you'll have another go, this time keeping in mind that I am merely a dilettant.

I don't know how many times I've shouted this from the mountain tops, but I KNOW that if I accelerate out into space inside an elevator that it will feel the same way as standing on the earth and things I let go of will drop etc. etc.! I've never thought of as geodesics as "straight lines" except in so far as the appear straight to those residing in the curved space and in so far as they are the shortest path between two points on a domed surface.

So here is what I've gotten from you so far. Spacetime curves are sometimes called tidal forces. Gravity, which I understand to be the pull exerted by a mass (what every particle does to every other particle in the universe), is not the result of moving through curved spacetime. Spacetime is merely a way to describe the phenomenon of being pulled towards a mass (don't quite know what that means). This pull that masses exert, which we call gravity, can exist without the presence of curved spacetime. Umm, do we happen to know of a planet or star or something out there which has gravity but does not curve spacetime?

 In GR there is no need for forces. In fact the basics of GR ar simpel and clear. The curved spacetime defines how particels move (read mass/energy) and mass/energy defines the curvature of spacetime.

 Originally posted by Peterdevis In GR there is no need for forces. In fact the basics of GR ar simpel and clear. The curved spacetime defines how particels move (read mass/energy) and mass/energy defines the curvature of spacetime.
The gravitational force still exists in GR. What Einstein did was to place the gravitational force as being equivalent to inertial forces which thus raised inertial forces to the level of "real" forces.

Particles moving through curved spacetime is not an explanation of gravity - it is merely a description of tidal effets in geometrical terms