# Confusion with curved space analogies

1. Jun 16, 2010

### DragonPetter

I've always been confused by the typical analogies I see when gravity as a space-time curvature is explained.

In 2-D it is usually a plane with field lines, and the surface of the plane is curved around an object. And so we are told a mass placed in this curvature will "fall" down the curve and that's how objects are moved in gravity. Its also given in the analogy of a funnel with a coin rolled down the funnel.

Then the more exact 3-D and 4-D models are given, but its still the idea that an object will "fall" along the curve towards the mass causing the curve.

My confusion is why do the objects move along the curvature? Say you took a stationary mass and placed it on the curved surface. Why would that mass want to move down the curve if its at rest? In the analogies, its actually gravity that is implied to move the object along the curve so they're using gravity to explain gravity which doesn't make sense.

So to sum it up, can someone answer or explain this question:
Take a stationary particle, of very low mass. Place it in the "curved" area of space-time around a large mass. What is making the particle accelerate on that curvature? Why isn't it happy staying in the spot its placed even though the spot might be curved in towards the larger mass?

2. Jun 16, 2010

### Ich

These analogies are misleading sometimes. I'm actually not sure in every case what they are supposed to show but here's what I think they should show.

You see a deforme plane. Now draw a straight line on it (or at least, try to do so). The standard procedure is to lay an arrow on the plane and move it always in the direction it is pointing, without deliberately rotating it. You end up with a curved line, which is calles a geodesic. That's what you can learn, in curved space geodesics ("supposed straight lines") are curved. Nothing to do with something falling in the funnel, it'd work exactly the same if the funnel were bent upwards.
Don't take it too literal as explaining gravity. There is one aspect missing that these pictures can't convey: It's not space that is bent, it's spacetime. The combination of space and time.
Every point in space, like a planet or so, is represented by a line in spacetime (its "worldline"). For example the world line of a particle at rest goes from past to furture, but not in a spatial direction. Like a straight line upwards in a spacetime diagram.
in curved spacetime, its worldline is curved into the space direction, which means that the particle is beginning to change its position due to gravity.

3. Jun 16, 2010

### Staff: Mentor

Consider two airplanes flying over the surface of the Earth. They start out at the equator, and fly northward with the same speed, along different longitude lines. Each plane flies a straight path, as far as it "knows," and continues to steer in the same direction. Nevertheless, as they go further north, they approach other, and collide at the North Pole.

4. Jun 16, 2010

### A.T.

The analogy you describe is one of curved space, and in fact it doesn't explain gravitational attraction of objects at rest in space. You need to include the time dimension to explain that:
http://www.relativitet.se/spacetime1.html
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html