# Question about the curvature of spacetime:

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1. Apr 30, 2013

### VeryConfusedP

There's something very fundamental about the curved structure of spacetime that is confusing me. Einstein is saying that gravity can bend starlight. In other words, if I have this right, a star's light will follow the curvatures of spacetime created by a large body of mass, like the sun.

Here's where it's getting very hard for me to visualize things. If I hop on a huge and powerful spaceship and go extremely fast towards this curvature of spacetime created by the sun's mass (exactly where the starlight is being bent), why can't my spaceship go right through that curved spacetime? I know this is an elementary way of putting it, but it's almost like we're going to run into this spacetime wall and bounce off it. (I know that's wrong, but I'm just trying to express how I'm visualizing this.)

I can see where massless light follows these curves, but why couldn't a fast-moving vessel restructure the curvatures of spacetime and not follow the same path as the light?

If you need clarification on my question, please let me know.

Last edited: Apr 30, 2013
2. Apr 30, 2013

### phinds

Judging from the question, I'd say you think the curvature is far more than it really is. It's not like photons are following some extreme curve that a spaceship could just plow through --- they would both be following a VERY gentle curve. Also, the photon is on a ballistic trajectory whereas the spaceship could fire it engines and then NOT be following the same curve as a ballistic trajectory.

Now, if you are talking about the near environs of a black hole, the curvature would not be so gentle, but the same comments as above still hold. Anything on a ballistic trajectory follows spacetime curvature and anything with the ability to be non-ballistic could follow a different path.

3. Apr 30, 2013

### Staff: Mentor

Curvature means that the geometry is not Euclidean. So if you make a triangle with 3 points connected by 3 "straight lines" (aka geodesics) then you will find that the angles add up to more or less than 180º. For example, on a sphere the geodesics are great circles, and the triangle formed by the north pole, the intersection of the prime meridian and the equator, and the intersection of the 90º longitude line and the equator, would have 270º.

Does that help?

4. Apr 30, 2013

### VeryConfusedP

DaleSpam:

Honestly, no, it doesn't--yet. But trust me, it's not you; it's me. I'm going to keep re-reading what you wrote until it all clicks.

Question: if something is moving on a straight line and there is no gravity present, is that straight line still called a geodesic? I thought geodesics were only present when gravity was present.

Thanks.

5. Apr 30, 2013

### HallsofIvy

Staff Emeritus
The main reason a space ship could not "go right through that curved spacetime" is that it gets really really hot!
I don't know where you got the idea that "we're going to run into this spacetime wall and bounce off it." That wouldn't happen at all. The center of the sun is still thin enough that, if it were not for the heat, we could pretty much go "right through it".

If would be different if you were trying to go through the earth's center or moon's center. But that is not a matter just of a gravitational field.

6. Apr 30, 2013

### Staff: Mentor

Is this correct? The Solar Center at Stanford says the density at the center of the Sun is 160 times that of water, or about 20 times the density of steel.

7. Apr 30, 2013

### Staff: Mentor

That doesn't seem like a correct visualization even for the light. Light just bends around a massive object; it doesn't bounce. And the bending is because the light is passing by tangentially; if the light moves radially inward towards the massive object, it won't bend, it will just go right in (the light will be blueshifted as it falls, but that doesn't change its trajectory).

8. Apr 30, 2013

### Staff: Mentor

Yes. A straight line is a geodesic.

In a flat space (sum of angles of any triangle = 180º) all the geodesics are straight lines and all straight lines are geodesics.

On a sphere (sum of angles of any triangle > 180º) all the geodesics are great circles, which locally look like straight lines but since they wrap all around the space usually you call them geodesics instead of straight lines. On a sphere there are no globally straight lines, but great circles are as close as you can get since they never turn locally, and it is only globally that they are not straight.

9. Apr 30, 2013

### pervect

Staff Emeritus
I hope this doesn't confuse the OP more, but there are a couple of types of geodesics. If you take a slice of space-time, that slice of space without time will have geodesics, which will represent locally "the shortest distance between two points".

Distances on a sphere such as the surface of the earth are a good example of a spatial geodesic.

One can also talk about space-time geodesics, which are in the four dimensions of space-time. Examples of these are the worldline of a spaceship or other object in free fall. In this example, the space-time geodesic will essentially maximize the time experienced by observers on the spaceship (the proper time) compared to any other path through space-time that starts and ends at the same two points.

(THat's slightly oversimplified, a more careful explanation would be that the space-time geodesic "extremizes" the time. If you don't understand the difference, don't sweat it overmuch).

Other examples: the worldlines of light beams are space-time geodesics. THere isn't any "proper time" to maximize or minimize, but the principles of optics tell us that it extremizes the "optical path length".

10. Apr 30, 2013

### WannabeNewton

The concept of a geodesic has nothing to do with gravity. It is a special type of curve defined in differential geometry. Specializing to GR, we say that particles follow geodesics if they are in free fall. If there are other forces present then they will no longer follow geodesics.

11. Apr 30, 2013

### VeryConfusedP

Newton:

How can there be curves (geodesics) in spacetime for objects to follow if there's no gravity? Far, far removed from a gravitational field, will a material object be at rest or "move" in a straight line, or will it "move" along a geodesic?

Edit: I just re-read Dale's post. He says that all straight lines are geodesics and vice-versa. I'll have to do some more research, because something isn't clicking right now. If it's a straight line, I don't know why we need to call it a geodesic instead of a straight line. It seems like that's just adding extra words to the dictionary to confuse people like myself even more.

12. Apr 30, 2013

### WannabeNewton

Well my point was that the concept of a geodesic itself has nothing to do with gravity; it is a purely mathematical construct. In GR we associate gravity with curvature of space-time and talk about freely falling particles as ones following the geodesics of the curved space-time (such particles are locally inertial).

If we are in flat space-time, the geodesic equation for e.g. massive particles is just $\frac{\mathrm{d} ^{2}x^{\mu}}{\mathrm{d} \tau^{2}} = 0$ where $\tau$ is the proper time. Remember that straight lines are geodesics too. A geodesic is a generalization of a straight line to curved manifolds.

13. Apr 30, 2013

### VeryConfusedP

Here's probably what I should have done. This picture is what's confusing me: http://en.wikipedia.org/wiki/File:Cassini-science-br.jpg

The light(the green stuff being emitted by that satellite) is following the curvatures of spacetime determined by the sun's mass. When I look at those blue lines (the curvatures) I want to know if I, on a spaceship and going really fast, can bust through those blue lines and create curvatures that the rays of light following those curvatures could not follow.

I'm just wondering if the curvatures that the light (green stuff) is following can be penetrated.

Last edited: Apr 30, 2013
14. Apr 30, 2013

### Staff: Mentor

This statement is true, but it's important to understand that the curvature of spacetime is *not* what that diagram is showing you. It is showing you the curvature of *space* around the Sun--but that curvature of space depends on a particular way of splitting up spacetime into space and time. (Also, the curvature of space alone does not completely account for the bending of light by the Sun; only a full spacetime calculation does. So the diagram is somewhat misleading since it gives the impression that the light traveling through curved space *does* completely account for the bending.)

No, they can't. The curvature of spacetime is the same for all observers, regardless of how they are moving. The curvature of space could be different for different observers, if they choose to split up spacetime into space and time in different ways; different states of motion might have different "natural" ways of doing the splitting. But once you've picked a way of splitting spacetime up into space and time, the curvature of space is also the same for all observers that use that way of splitting.

I understand the intuitive appeal of the idea that we could somehow find a way to "cut through the curvature", the way tunneling under the Earth's surface can shorten our path to a distant location. But you can't "tunnel underneath" the curvature of spacetime; there's no external "space" to tunnel into. Spacetime is all the space and time there is.

15. Apr 30, 2013

### Staff: Mentor

Careful. I said: "In a flat space (sum of angles of any triangle = 180º) all the geodesics are straight lines and all straight lines are geodesics." The reason we use the word "geodesic" is because we want to generalize the idea of a straight line to include curved spaces as well as flat spaces. Geodesics are the general case, straight lines are the specific case only for flat spaces.

All straight lines are geodesics, but not all geodesics are straight lines. This is because some spaces are curved so that there are no straight lines. For example, on a sphere, the geodesics are great circles. There are no straight lines on a sphere, but a great circle serves the same purpose on a sphere that a straight line does in a flat space, specifically it minimizes the distance between two points.

16. Apr 30, 2013

### Staff: Mentor

Completely dismiss this picture from your mind. There is so little value to it and so much wrong with it that it is not worth even worrying about. This picture does NOT show how gravity works in GR.

17. Apr 30, 2013

### Staff: Mentor

VeryConfusedP is a sockpuppet of a banned member and is now gone.

18. May 4, 2013

### Staff: Mentor

Imagine that you are a 2D being living within the curved 2D surface of a globe. You are unaware that the radial direction even exists, and are physically unable to move out of the surface of the globe. If you could move out of the surface, you could go directly from New York to Melbourne by moving across inside the globe. But this is not possible for you. The best you can do is to go from New York to Melbourne by following a great circle. The great circle is a geodesic for the 2D space that you are trapped in. No matter how fast you go, you can't escape the surface of the sphere.