Would gravity still act if all objects in a closed system somehow became stationary?

Question

I have been trying to understand the implications of general relativity. I unfortunately don't have a good knowledge of advanced topics and I may have made some silly assumptions.

As far as I understand, spacetime dictates the trajectory of an object, and the object curves spacetime. Objects follow the shortest path, and it appears as if things are being pulled when instead they're just accelerating in a specific way due to the curvature. Gravity is a fictitious force.

I'm confused about what would happen if we imagine a universe with two identical stationary objects. I'm guessing that because it's not actually possible for anything to be completely stationary (because we cannot reach absolute zero (uncertainty principle?)), these objects will move along the curvature. But if we consider it was possible for objects to be completely stationary, does this mean that these objects won't follow the curvature since they're not moving to begin with and it would appear as if gravity has stopped working. The objects stay stationary instead of crashing into each other.

What would happen if there are two identical stationary objects and I apply a force to one of the objects, such that the direction of the force is perpendicular to the line connecting the two objects? I'm guessing that it should start to orbit the other object, but I also know that in this inertial frame of reference, since there are two objects, I shouldn't be able to tell who's moving. So the outcome of some force being applied should be symmetrical, so does this mean the objects would start to chase each other?

But then, in a situation where the objects are not identical, if I move the heavier object, would it still appear as if the smaller object has started orbiting due to a force pulling it? But this sounds like movement in one stationary object has induced movement in another stationary object (assuming stationary objects were possible)?

Answer 1

By 'stationary', you would mean 'stationary with respect to the spatial axes of a certain inertial frame of reference'. However, a stationary object would still 'move' along the time axis -- in fact, it will 'move' as fast as it can (=at the speed of light) along the time axis, if it is stationary spatially.

The thought is that your four-dimensional 'speed' is always constant. The only difference between a moving body and a stationary one is the composition of their four-dimensional 'speed'; if it's stationary, then all of its '4D speed' will come from its temporal 'motion'; if it's not, then the 'speed' will be a combination of its spatial motion and temporal 'motion'.

Of course, here the words 'move' and 'speed' should be interpreted as somewhat metaphorically, for you cannot really make sense of things moving along the temporal axis. (or maybe you could.)

Anyhow, from the four-dimensional point of view, you're always 'moving' at a constant 'speed', which is the speed of light, regardless of your spatial motion or temperature or whatever.

(By the way, all you need to articulate this is just special relativity: Suppose that it took t seconds for an object S moved from a point A to a point B at a constant speed v in a certain inertial frame K. In the frame of reference of S (i.e., the frame of reference where s is always at the origin), let's say, the same journey took t' seconds.

Then we could think of $\frac{t'}{t}$as the 'temporal speed' of the object S relative to the frame K. It tells you, so to speak, how slow S's clock ticks with respect to a clock in K.

Then there's the following special relativistic relation between S's 'temporal speed' and its 'spatial speed (v)':

$(\frac{t'}{t})^2 + (\frac{v}{c})^2 = 1$

This is what I meant when I said your '4D speed' is always constant.

You always follow some spacetime trajectory regardless of your velocity with respect to a certain frame. So, yes, gravity still works even in the physically impossible hypothetical situation where things have zero spatial motion with respect to a certain inertial frame.

2. Regarding the indiscernible objects: if you apply force to one of the objects but not to the other, then, of course, you can tell which one is which; one that experiences acceleration is the one that you pushed, and whether something is accelerated or not is not a relative matter in general relativity. What is completely relative is inertial motion (in SR) or geodesic motion (in GR). If a bucket of water is rotating, then everyone should agree that it's rotating, for angular motion is a form of acceleration and acceleration is not totally relative.

Answer 2

To restate your question:

Will there be gravity when two objects are stationary ?

• What is a stationary object?

1. As rightly pointed out above, the object with mass can be stationary only in space but not in time. Everything moves in time except for photons! As photons follows null geodesics (Null Geodesic) hence they do not experience time!

2. You have quoted that objects can not be truly stationary because of the 'Uncertainty Principle'. This is only true for subatomic particles. For larger objects these effects are very negligible e.g. Earth's position uncertainty can be calculated via De Broglie's equation as λ = 3.68x10^-63 m, which is smaller than the size of a nucleus of an atom (~10^-15 m).

3. For our case lets assume we are dealing with objects big enough to neglect uncertainty principle. Hence, they can be stationary in space. Now, lets address if they will collide with each other.

• Will they experience gravity?

1. According to Newtonian mechanics, two stationary objects separated at an asymptotically infinite distance experience an attraction, and will eventually come towards each other.
2. According to general relativity, the same outcome is expected. In general relativity the matter curves spacetime, and spacetime dictates the motion of the matter. The matter takes the shortest path in the spacetime to move. These paths are straight lines on a Euclidean spacetime (a spacetime with zero curvature). However, this is not the case in a general curved spacetime. An apple on a table looks stationary, but it is moving in its geodesic path (radially towards the center of Earth) only to be prevented by the table. Whereas, a free-falling Apple is following the geodesic, which is pointed radially towards the center of Earth.
3. A observer can always take a free-falling frame of reference along the free-falling Apple, and make it look stationary in this frame of reference! This is possible as a free-falling frame of reference (accelerated frame) is a inertial frame of reference in general relativity unlike in Newtonian mechanics.

Conclusion: The two stationary objects with mass/energy will distort the spacetime. Hence, the two objects will start moving in their respective geodesics to eventually collide into each other. Therefore, it is essentially like an Apple falling onto Earth. The geodesic curve is worked out in this post by John Rennie.