Curvature of Spacetime: an inacurate Description?

[Red Fox]

I think about general relativity often, specifically about the curvature of spacetime in the presence of matter (gravity). For a while, I understood much of this concept, but certain things escaped me: when objects are moving, it is easy to see how curved space causes matter to move in the way it does; however, it is difficult for me to understand why an object that is in relative state of "rest" would move if another stationary mass was introduced. Then I realized that thinking of matter's effect on space time as simply a "curvature" may not be an entirely accurate analogy. For me, "curved" space implies only that what would be a straight line or path in space is now bent to some degree. If this were the case, i would only make sense for an object to be effected if it is moving relative to the mass causing the curvature, but as we know, gravity effects matter in all forms of relative motion. If you change the description from curvature to distortion, or some other, more suitable adjective, it would describe and relate to what we observe much better. I am not by any means an expert in this area, and i would like to know what others thoughts on my little pseudo approach to this concept are.

 

[CompuChip]

So maybe the choice of words is what confuses you here.
Would it be less confusing to you to say that the spacetime metric depends on the matter content of the spacetime (and vice versa)?

 

[A.T.]

when objects are moving, it is easy to see how curved space causes matter to move in the way it does;

Curved spacetime, not just space.

however, it is difficult for me to understand why an object that is in relative state of "rest" would move if another stationary mass was introduced.

In spacetime, everything "moves". You can be at rest in space, but then you are still advancing trough time, and therefore trough spacetime.

Try chapter 2 of this:
http://www.relativitet.se/Webtheses/tes.pdf

 

[DrGreg]

As A.T. said, it's curved spacetime, rather than curved space.

Any point object in space is represented in spacetime by a line (a "worldline") regardless of whether you think it is "moving" or "at rest".

The analogy to think of is the curved, two-dimensional surface of the Earth. If two people are on the equator and each heads north in straight line, at first they seem to be traveling on parallel lines. But as they approach the North Pole it becomes clear their paths aren't parallel at all, they are moving closer together. If the Earth were flat, we couldn't explain that. But we can explain it as due to curvature of the Earth's surface. There are mathematical equations that can describe this curvature purely in terms of the two dimensions of the Earth's surface.

Similarly, in spacetime, two objects can begin with apparently parallel worldlines (which means they are a constant distance apart) but later the worldlines are getting closer together, i.e. the objects move towards each other. The mathematical equations that describe this are very similar in structure to the equations of spherical geometry of the Earth's surface, so by analogy we call this "curvature" too.

 

[Naty1]

however, it is difficult for me to understand why an object that is in relative state of "rest" would move if another stationary mass was introduced.

It's not intuitive: if it were, it would have been "discovered" thousands of years before Einstein.

You might think in terms of the "rubber sheet" analogy...when a second mass is added the whole sheet sinks a bit more, and a prior rest mass and the newly introduced mass would tend to "sink" (move) towards each other...like when somebody sits on a mattress right next to you...you can feel the additional depression...

 

[LURCH]

...like when somebody sits on a mattress right next to you...you can feel the additional depression...

I used to have that problem, 'till I met my wife.

 

[A.T.]

You might think in terms of the "rubber sheet" analogy...when a second mass is added the whole sheet sinks a bit more, and a prior rest mass and the newly introduced mass would tend to "sink" (move) towards each other...like when somebody sits on a mattress right next to you...you can feel the additional depression...

This doesn't answer the OP's question, why the curvature of space should in any way affect objects at rest in space. In fact it doesn't. The key is to consider also the time dimension. It's curvature of spacetime.

Where is the time dimension on the mattress/rubber sheet? It only represents two space dimensions. The only reason why the depression in the mattress affects you, is gravity. You are trying to explain gravity by gravity.

Free falling objects in GR are not moving into depressions in curved space. They are moving as straight as possible trough curved spacetime.

Here an interactive diagram visualizing this:
http://www.adamtoons.de/physics/gravitation.swf

 

[Red Fox]

Thanks, I think I understand it a bit more now.

 

[Red Fox]

A.T., thank you. Your answer perfectly suites my problem. When i was thinking of spacetime, and its curvature, I was thinking in terms of the bowling ball on the sheet analogy in exactly that flawed way: using gravity to explain gravity. I hadn't figured time into spacetime properly. When i examined your link, it was one of those "of course!" moments. Even though i thought i was thinking in terms of space and time as one, i was really still thinking of them separately. Again, thank you.

 

[A.T.]

A.T., thank you.

You're welcome!

When i was thinking of spacetime, and its curvature, I was thinking in terms of the bowling ball on the sheet analogy in exactly that flawed way: using gravity to explain gravity.

Yeah, the stupid sheet analogy used everywhere to explain gravitation in relativity, did never make sense to me neither. Then I read yet another book on relativity, and the first picture in the gravitation chapter was that rubber sheet & bowling ball again. But then I read the caption of the picture: "Popular but completely wrong analogy. Better forget it now!". That book was "Relativity Visualized" by Lewis Carroll Epstein. Here are some pictures from it explaining the different effects of curved time and curved space:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

 

[feynmann]

when objects are moving, it is easy to see how curved space causes matter to move in the way it does;

All of Newtonian gravitation is simply the curvature of time. The space of curved spacetime will not causes matter to move in the way it does, only the curved time will.
http://www.gravityfromthegroundup.org/pdf/timecurves.pdf

 

[mtworkowski@o]

why is the concept of spacetime more useful than the concept of gravity? I don't think that's what i want to ask. My question concerns the path of sun light during solar eclipse. How did that prove that Einstein's theory about curved space is right and just plane old gravity bending light couldn't be right? I don't have a degree in physics or anything, so if anyone wants to answer the question i'd be happy if i could understand the answer.
thank you.

 

[jtbell]

My question concerns the path of sun light during solar eclipse. How did that prove that Einstein's theory about curved space is right and just plane old gravity bending light couldn't be right?

Both Newtonian gravitational theory and Einstein's general relativity predict that a ray of light "bends" as it passes the sun, but they predict different amounts of bending. The observed amount of bending agrees with general relativity, but not with Newtonian gravitation.

 

[Dale]

Also, GR explains why inertial and gravitational mass are the same. It is just left as an odd coincidence in Newtonian gravity.

 

[A.T.]

My question concerns the path of sun light during solar eclipse. How did that prove that Einstein's theory about curved space is right and just plane old gravity bending light couldn't be right?.

Initially Einstein modeled Newtonian acceleration (that would also bend light) as http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_time.gif", which doubles the amount of light bending and matches the observed value. It also explains the observed orbit precession.

 

[feynmann]

How is spacetime curved if the present gravity field is completely uniform and there are no tidal forces. Clocks at a same height would tick the same, at different heights (to the gravity source) would tick differently. But what about space? How is space curved in the absence of tidal forces?

Often curvature is introduced with falling elevators without tidal forces. The observer in a falling elevator sees a light ray going from on side of the elevator wall to the other as a straight line. An outside observer sees a bended line. Thus gravity bends spacetime they say. Later then tidal forces and the non-uniformity of gravity fields is mentioned and made responsible for curvature.

So again my question: What curvature of spacetime describe? Newtonian gravity or tidal forces of gravitational field? Thanks...

 

[N721YG]

I would also like an expert opinion on this but am not sure of your actual question. I understand gravity curves spacetime. On earth, gravity from the Sun and moon cause the tidal forces. You do not need tidal forces to bend space time.

 

[JustinLevy]

In such introductions they are using the word "curved" to mean that lines which are straight in your coordinate system are NOT geodesics. So while the beams of light in an accelerated frame "curve" compared to the coordinate axes, the point is that the light is actually following a geodesic of spacetime.

You are correct though that the intrinsic curvature of such a frame is actually zero. An accelerated frame in flat spacetime will show the same thing (and yes, with no tidal forces). These introductions are just to help you start imagining in non-Euclidean geometry (even though you used cartesian coordinates, the axes are not necessarily "straight lines" geometrically). Once this kind of thinking is introduced, then they move onto the more complicated non-Euclidean geometries seen in actual curved spacetime with matter.

Does that make sense?

EDIT: For context, I responded to Feynmann's openning posting before his thread was merged into this. So do not take this as responding to anything else previously in the merged thread.