# Mass causes curvature?

1. Jul 7, 2006

### blumfeld0

I need help in trying to explain to someone how gravity creates curvature and paths of particles are curved around this curvauture?
like the sun on a trampoline and the earth's path around it being bent as it orbits the sun. so why does the sun bend the space around it? and why does the earth travel around this bent space? i am looking for a geometrical explanation. thank you

blumfeld0

2. Jul 8, 2006

### pervect

Staff Emeritus
"Why" questions don't really have answers. If you understand that the Earth orbital motion, which appears "curved", can also appear to be a straight line in curved space-time, you've got the basics.

Note that most of the "curvature" in space-time can be attributed to time, and not space. This is one of several areaa where the trampoline model isn't terribly good. The trampoline model invites one to think of space as curved, because it's easier to visulaze. The actual math of relativity is that space-time is curved rather than just space being curved. Furthermore the part of curvature that's most significant is due to time, not space.

We can actually measure this curvature by the facts that clocks tick more slowly the deeper they are in a gravity well.

It is possible, using the principle that clocks tend to move in such a manner as to xtremize proper time, to work out the orbits of slowly moving bodies.

Very fast moving bodies have to take into account spatial curvature too, but slowly moving bodies can be modelled very well by looking only at the time part of the curvature, namely gravitational time dilation.

Enough physics to appreciate "the principle of least action" helps a lot, unfortunately to get to this level rigorously requires a fair amount of math (calculus and calculus of variations).

3. Jul 8, 2006

### Chaos' lil bro Order

I think when you consider that gravity, energy and matter are all equivalent, you can see that any of these, in a sufficient amount, will appreciably curve space. Density is the true key to curving space. A 1kg book will not appreciably curve space, but put that 1kg in a space confined to 1 nanometer squared and space will certainly curve appreciably.

4. Jul 8, 2006

### MeJennifer

Here is the how and why:

Basically you have to start with the principle of relativity and the fact that the speed of light is constant in each frame of reference. One of the implications of this principle is that particles observe a contraction of the distance of approacing particles. Sometimes called Lorentz or length contraction.
As long as these particles approach with a constant rate we can use a linear calculation to determine the amount of contraction.
Basically this phenomenon of contraction can be modeled adequately in a flat 4-dimensional Minkowski space.
However once the rate of approach is no longer constant the calculation is no longer linear and we can no longer model this in a flat Minkowski space. In order for the model to work we have to allow for curvature in this 4-dimensional Minkowski space.
So we need a 4-dimensional Minkowski space with curvature to provide a framework that caters for the principle of relativity and acceleration.

Finally there is Einstein's principle of equivalence. What he is saying is that acceleration and gravity are equivalent.
Hence mass must curve the 4-dimensional Minkowski space just like acceleration does.

The last step is going from the model, the 4-dimensional curved Minkowski space to reality.
This space is an adequate representation for calculations of classical mechanics. But is the universe really a 4-dimensional Minkowski space with curvature? That of course we cannot prove, and surely the string theory advocates would prefer to add a couple of dimensions to that.

But that is basically where the curvature thing is derived from.

Last edited: Jul 8, 2006
5. Jul 9, 2006

### pmb_phy

That isn't quite right. The weak form of the equivalence principle states that a uniform gravitational field is equivalent to a uniformly accelerating frame of reference. The strong form of the equivalence principle states that locally a gravitational field can be transformed away at point in spacetime (i.e. at any event). The strong form has also been stated in other ways. But in general a gravitational field is not the same as acceleration. That was never how it was stated by Einstein.
This does not follow from the equivalence principle. If there is matter at an event E in spacetime then the spacetime curvature at that event is non-zero. However the spacetime near that event can indeed vanish, depending on the exact field of course. Gravitational acceleration is not the same thing as spacetime curvature. Gravitational tidal forces are the same thing. Where there are tidal forces there is spacetime curvature and where there is spacetime curvature there are tidal forces. Curvature has an absolute existance and cannot be transformed away like the gravitational field can be.

Pete

6. Jul 9, 2006

### DaveC426913

This is a bit misleading - it's like saying matches (or other fuel) and flames are equivalent, and that enough of either can start a forest fire. Yet you cannot have a flame without a combusting fuel.

You see, GR demonstrates that gravity is not a thing at all, it is nothing but an effect. Mass or energy density creates curvature in spacetime. Period. Gravity is nothing more than our perception and measurement of that curvature.

You can't have a bunch of gravity all by itself any more than you can have a bunch of flames without a fuel source.

7. Jul 10, 2006

### Garth

And yet gravitational waves carry off energy......

Garth

8. Jul 10, 2006

### Farsight

Blumfeld: I'd like to back up Pervect and Dave. IMHO gravity isn't a thing that creates "curvature", the underlying mass affects spacetime like Pervect was saying. A simple analogy is that you've got two legs, but time runs slower in your left leg than your right leg, so you find yourself walking round in circles. You give this effect the name "gravity".

Garth: I don't know if it's relevant, but surely you can't have energy all by itself either.

9. Jul 10, 2006

### pmb_phy

The relationship between gravitation and curvature is really quite simple. First of all GR does not explain gravity. It is a general theory of relativity. This means that it covers all kinds of coordinate systems rather than simply Lorentz coordinates (i.e. inertial frames). One can, as Einstein put it, "produce" a graavitational field by a change in spacetime coordinates. However this does not mean that one can create spacetime curvature by changing the coordinate system.

GR describes the exact same phenomena (gravitational acceleration and tidal acceleration) that Newton's theory describes. What is refered to in Newtonian physics as "tidal forces" is now called "spacetime curvature" in Einstein's theory. The relative acceleration of two particles in Newton's theory is describe in GR by two geodesics which deviate.

It is to be noted that according to Einstein, gravity and curvature are not identical phenomena. In Newtonian gravity one can have a gravitational field with no tidal forces (i.e. a uniform gravitational field). The spacetime curvature of such a field is zero. Recall Newton's law of gravitation in differential form. At a point r in space the Laplacian of the gravitational potential at r is proportional to the mass density at r. Howevever at positions outside the clump of mass which generates the field the mass density is zero. The tidal forces at such locations may or may not be zero. An example of such a field is that of a spherical cavity cut out of a sphere of uniform mass density. When the center of the cavity is displaced from the geometrical center of the sphere a uniform gravitational field will be found within the cavity. Likewise, Einstein's tensor, which kind of replaces the Laplacian, is proportional to the stress-energy-momentum tensor, i.e. that object which completely describes mass. Where there is matter there is spacetime curvature. The converse need not be true though.

Pete

10. Jul 10, 2006

### DaveC426913

Let me point out a different analogy then.

It is like saying water and (water) waves are equivalent.

You cannot have (water) waves without the water in which they do their thing. The waves are simply an effect - a description - of the motion of the water.

Gravity is simply a description of the curvature of spacetime. Gravity, in and of itself, is not what causes the Earth to revolve about the Sun.

11. Jul 10, 2006

### pmb_phy

Not as the term "gravity" was used Einstein. See above for clarification.

Pete

12. Jul 10, 2006

### blumfeld0

Hello. Thank you for your nice responses

""Gravity is simply a description of the curvature of spacetime. Gravity, in and of itself, is not what causes the Earth to revolve about the Sun.""

so why does the earth go around the sun according to einstein?

thanks

13. Jul 10, 2006

### pervect

Staff Emeritus
There are really a couple of different analogies that are popular, and not directly related, with respect to gravity.

The first is the famous "elevator" thought experiment.

The second is the idea that bodies, in the absence of any forces, follow the equivalent of straight lines, known as geodesic, in a curved space-time.

These ideas are not incompatible, but they are different descriptions of the same thing.

I get the impression from your original question that you are more interested in the second idea.

A very brief sketch of this is the famous "ants crawling around on an apple". The ants each follow as straight a path as they can. However, while they start out moving away from each other, they eventually start approaching each other again. One could attribute this motion to a "force of attraction between the ants", but as far as they are concerned, they are just following straight lines.

http://homepage.mac.com/stevepur/physics/riding/Riding_session_1.pdf [Broken]

as a reasonable example of this sort of explanation. There are some very good and detailed explanations of this sort in MTW's famous textbook "Gravitation" as well, however this book as a whole is not written on a popular level.

"Curvature", as formally defined, turns out not to play any significant role in the elevator gerdankenexperiment. Thus when you mix and match the different examples, you invite a certain amount of confusion if one adheres to all the technicalities. It's the physics equivalent of "mixed metaphors" in Engish text.

BTW, I wouldn't particulary hang onto Einstein as the best and only source of information on gravity. In some ways, Einstein's original views are a bit cumbersome. I think it is better to focus on the physics, and less on the personalities, myself.

Last edited by a moderator: May 2, 2017
14. Jul 10, 2006

### MeJennifer

Well, so are you saying that clocks don't run slower there?

15. Jul 11, 2006

### pmb_phy

No. However the gravitational redshift of light can be detected in a uniform gravitational field. In fact this was the first field in which the phenomena was calculated.

Pete

16. Jul 11, 2006

### MeJennifer

So you are saying that gravitational reshift has nothing to do with a slower clock?

Let me get this right, clocks don't run slower AND there is gravitational reshift, and the redshift is perfectly explainable with a flat space-time.

Perhaps you could explain this, to me it does not make any sense whatsoever.

Last edited: Jul 11, 2006
17. Jul 11, 2006

### pervect

Staff Emeritus
If you have two clocks, one higher than the other, in the gravitational field of the Earth, you'll see a gravitational redshfit of the lower clock from the POV of the upper clock.

If you have two clocks, one higher than the other, in the "gravitational field" of an accelerating elevator, you'll see exactly the same effect.

But if you look at the pair of clocks in the accelerating elevator from the POV of an inertial frame (rather than from the POV of the clocks), you won't see any gravitational redshfit at all. Instead, you'll see a perfectly inertial coordinate system, along with some doppler shifts due strictly to velocity.

18. Jul 11, 2006

### RandallB

This does not ring true for me.
How can the POV of an inertial frame expect to see an inertial coordinate system for the source of light it observes from the elevator?
It must see an accelerating coordinate system with Doppler shifts affected by an increasing relative velocity, not a fixed velocity.

19. Jul 11, 2006

### MeJennifer

For the good order it seems this is what is claimed:

In a uniform gravitational field the following is true:

- The space-time is completely flat
- The clocks do not run slower
- There is gravitational redshift

To me that sounds like a contradiction.

How could one possibly explain gravitational redshift, in a flat space and claiming there is no time dilation?

20. Jul 11, 2006

### pervect

Staff Emeritus
I think some of the statements complained about by MeJennifer are mutually contradictory too. Here's the situation as I would describe it.

A "uniform gravitational field" is described by coordinates which have an associated metric of

-(1+gz)^2 dt^2 + dx^2 + dy^2 + dz^2

1) The above space-time is completely flat, in the sense that the Riemann curvature tensor for the above metric is zero.

Other loose senses of what the term"flat" means are occasionally used, it's not really a very precise term. In the precise sense of having a zero Riemann curvature, the above metric is definitely flat, however.

This "flatness" can also be seen by the fact that there is a mapping via a change of variables to a flat-space metric.

-dtt^2 + dxx^2 + dyy^2 + dzz^2

This illustrates flatness because the Riemann curvature tensor for the above metric (the flat Minkowski metric) is widely known to be zero, and because a tensor that is zero in one coordinate system is zero in all coordinate systems.

The choice of coordinates that puts the metric in the above form represents an inertial coordinate system. (tt,xx,yy,zz).

Note that the choice of coordinates is completely arbitrary - it's a human choice.

It can easily (hopefully) seen, that for an object "at rest" in the inertial coordinates, dxx=dyy=dzz=0, and thus dtau = dtt, and there is no time dilation. Coordinate time is the same as proper time.

It can also be seen that for an object "at rest" in the accelerated coordinates, dx=dy=dz=0, and thus dtau = (1+gz)*dt, and there is gravitational time dilation. The relationship between coordinate time and proper time is a function of "height", the z-coordinate. This is time dilation, when coordinate time is not the same as proper time, and the amount of time dilation depends on the "height" of the object, i.e. it's z coordinate.

It should be understood that the object with xx=yy=zz=constant, which is "at rest" in inertial coordinates, is not following the same trajectory as the object with x=y=z=constant, which is "at rest" in the coordinate system used by the accelerated observer.

It should also be understood that a lot of the confusion arises from the use of concepts such as 'time dilation' that are coordinate dependent. When one chooses to use a coordinate dependent explanation (because of familiarity on the part of readers, usually), one has to live with the fact that it is coordinate dependent.

An explanation formulated purely in terms of coordinate independent quantites (like the Lorentz interval and abstract tensor notation) doesn't have to deal with these issues. But it is not generally as accessible.

Let me add a note on "curvature causes gravity". This is still a valuable notion, it comes directly from Einstein's equation

G_uv = 8 Pi T_uv

Here G_uv is a tensor, called the Einstein tensor, which is dervied from the Riemann tensor, which represents the curvature of space-time. If the Riemann tensor is zero, the Einstein tensor mus also be zero.

T_uv is another tensor, which represents the density of energy and momentum per unit volume.

Note what the equations says about the elevator experiment:

The equation says that there is no appreciable mass in the elevator experiment (just some small test masses, i.e. we are ignoring the gravity generated by the elevator car itself).

It also says that there is no Einstein curvature of the space-time in the elevator. We can go further than this - not only is there no Einstein curvature of the space-time in the elevator, there is no Riemann curvature either.

So Einstein's equation tells us that matter curves space-time, and it also tells us that in the elevator experiment, because there is no matter (or rather, because there is a negligible amount of matter, not worth worrying about), there is no curvature.

The difficulty is that people expect G_uv to represent "felt" gravity, but it doesn't. It's a measure of the curvature of space-time.

The common idea of "felt" gravity is basically the magnitude of the 4-acceleration of a particular observer. If the observer is stationary, this can be equated to the Christoffel symbols.

This differs also from what Pete says that Einstein likes to call gravity, which are the metric coefficients.

This may be a bit confusing but the bottom line is that metric coefficients, curvature tensors, and the magnitude of 4-accelerations are all precise things defined by physics, and all of them are relevant to the topic of gravity. It is the responsibility of the reader to make an informed decision about which of the above best matches their own idea of what "gravity" really is. I would suggest that the last quantity, the magnitude of a 4-acceleration, is what most people have in mind when they talk about "gravity". That's the force that one feels on the seat of one's pants when one sits down in a chair.

Last edited: Jul 11, 2006