# Stupid question

As always I come to this message board with a stupid question in hopes of being educated.

Someone brought this up and I had no answer.

Gravity is an effect of warped space, is it not?
The more massive an object, the more warped the space is around it.

Less massive objects travelling near a more massive object will attempt to move in a straight line, but since space is warped, the trajectory will curve.

But why will otherwise motionless objects (motionless wrt each other) gravitate toward one another?

## Answers and Replies

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Spacetime will be curved toward their center of mass.

JesseM
shrumeo said:
Less massive objects travelling near a more massive object will attempt to move in a straight line, but since space is warped, the trajectory will curve.
The closest thing to a straight line on a curved surface is a "geodesic". But in general relativity the geodesics that objects follow are not the path through curved space with the shortest length, but rather the path through curved spacetime with the greatest proper time (time as measured by a clock that follows that path). And you can't be "stationary" in spacetime, every object is following some path through spacetime (its worldline), even if it is stationary in space (with respect to some coordinate system).

pervect
Staff Emeritus
shrumeo said:
As always I come to this message board with a stupid question in hopes of being educated.

Someone brought this up and I had no answer.

Gravity is an effect of warped space, is it not?
warped space-time, not just space.

Every object "moves through time". So when you trace the geodesic through time, the warped space-time causes them to experience a relative acceleration.

Thanks for the replies. They all help.

They have made my concept of this less muddy, but I guess I need to ask a follow up.

So, although the two objects are stationary (wrt to each other) in space, they gravitate toward each other "because" time is passing? Can I say that?

Is the passage of time "why" nothing with mass can be "stationary" in space?

pervect
Staff Emeritus
shrumeo said:
Thanks for the replies. They all help.

They have made my concept of this less muddy, but I guess I need to ask a follow up.

So, although the two objects are stationary (wrt to each other) in space, they gravitate toward each other "because" time is passing? Can I say that?
Yes, though you may get some questions about why this is true. The detailed explanation needs some math, unfortunately - calculus to be specific. Also, it would be better if you identified one of the pair of masses as being large, distorting space-time, and the other mass as being small, a "test mass".

If we consider a small test-mass that is stationary in some coordinate system in curved space-time, with coordinates $x^0=t,x^1=x,x^2=y,x^3=z$ the force that must be acting on it to hold it stationary is given by the geodesic equation

$$\frac{d^2 x^i}{d \tau^2} + \Gamma^i{}_{jk} \left({\frac{dx^j}{d\tau}\right) \left( \frac{dx^k}{d\tau} \right) = F/m$$

where F is the force, and m is the mass of the object. The $\Gamma$ values are numbers which represent the Christoffel symbols of the curved or distorted space-time. This curvature or distortion is presumably caused by the larger mass, though there can in general be other causes such as the choice of a non-inertial coordinate system.

You can look up Christoffel symbols on the www, but their exact defintion is somewhat technical, just think of them as quantities that are all zero in normal flat space-time in an inertial coordinate system, and non-zero in distorted or non-inertial space-times.

Because the object is stationary, most of the terms in this equation will be zero, because $dx^1/d\tau=dx^2/d\tau=dx^3/d\tau=0$

The time derivative terms $dx^0/d\tau$ represent "motion through time", and can be regarded as the "cause" of the force on the object due to the curvature of space-time, a force (like gravity) that must be balanced by an external force in order for the object to remain stationary. There will be a force in the $x^i$ direction if and only if $\Gamma^i{}_{00}$ is non-zero.

Is the passage of time "why" nothing with mass can be "stationary" in space?
I wouldn't say this - it is possible for something with mass to be stationary in space (with repsect to some coordinate system), though it may require an external force. So I don't see how it makes sense to say that nothing with mass can be stationary in space.

It is of course necessary to specify a specific coordinate system when one is talking about an object being stationary - i.e. one must specify what an object is stationary with respect to.

A few good analogies to try to understand this without math:

All objects fallow the path of least resistance. If you set a ball on a flat surface, the easiest thing for the ball to do is stay still. This is analogous to an object motionless in space.

If you set a ball on a slanted surface the easiest thing for the ball to do is to roll down the surface at an exponential rate. This is similar to the path of an object falling toward the earth

If you were to roll a ball from east to west, across a surface that is slanted downward from north to south, the ball will start off rolling westward, but it’s path will curve until it’s rolling in a southwesterly direction. Eventually the ball will be rolling southward.

Another more popular analogy is to set a bowling ball on a waterbed. This will cause the surface of the waterbed to dip inward. It will create a curved (warped) surface that will have more of a slope the close to the bowling bass, and less of a slope further away from the bowling ball. This dip represents warped space/gravity created by the presence of the bowling ball.

If you set a marble far away from the bowling ball, it will stay put, unaffected by the bowling ball’s “gravity”. If you set a marble closer to the bowling ball, where the curve starts, the marble will roll straight toward the bowling ball. If you roll the marble so that it passes by the bowling ball, it will get caught in the curve of the bed and circle the bowling ball, before eventually colliding with it.

LosBoogie said:
A few good analogies to try to understand this without math:

All objects fallow the path of least resistance. If you set a ball on a flat surface, the easiest thing for the ball to do is stay still. This is analogous to an object motionless in space.

If you set a ball on a slanted surface the easiest thing for the ball to do is to roll down the surface at an exponential rate. This is similar to the path of an object falling toward the earth

If you were to roll a ball from east to west, across a surface that is slanted downward from north to south, the ball will start off rolling westward, but it’s path will curve until it’s rolling in a southwesterly direction. Eventually the ball will be rolling southward.

Another more popular analogy is to set a bowling ball on a waterbed. This will cause the surface of the waterbed to dip inward. It will create a curved (warped) surface that will have more of a slope the close to the bowling bass, and less of a slope further away from the bowling ball. This dip represents warped space/gravity created by the presence of the bowling ball.

If you set a marble far away from the bowling ball, it will stay put, unaffected by the bowling ball’s “gravity”. If you set a marble closer to the bowling ball, where the curve starts, the marble will roll straight toward the bowling ball. If you roll the marble so that it passes by the bowling ball, it will get caught in the curve of the bed and circle the bowling ball, before eventually colliding with it.
I've seen these types of analogies before, but they all rely on the presence of gravity itself, so I've never found them to be useful in actually understanding things.

and this one doesn't seem right:
All objects follow the path of least resistance. If you set a ball on a flat surface, the easiest thing for the ball to do is stay still. This is analogous to an object motionless in space.

An object in space can't really be motionless with respect to everything in the universe. So pretty much everything I can think of will be in some sort of motion with respect to at least something.