# Question on equivalence of acceleration and mass with respect to gravity

Layman’s question here. Let’s say I’m standing on the inside rim of a rotating space station spun at right rate to produce earth-like gravity. Does the spinning warp space time? If so, how can a small spinning spacecraft produce the same amount of curvature as a huge mass like the earth?

It can’t be equivalent, because an astronaut floating right outside the spacecraft would not be pulled down to it, but would be pulled down to the earth.

My understanding is that from my point of view standing on the rim, my acceleration and the dilation of the circumference of the station vs it’s radius produces the local experience of a curvature, and since curvature defines gravity, I feel gravity.

Can someone let me know if I’m correct? Thanks!

• I do not think that you must put in GR. I would forget curvature except for the fact that the spinning station would be probably curved indeed. Just carousel physics here. Commented Dec 29, 2018 at 9:44
• Spinning an object produces centripetal acceleration, but does not warp spacetime. You weigh more on the north and south pole then the equator by a bit due to spinning of earth. However, spacetime is uniform (mostly) around our earth. Commented Dec 29, 2018 at 12:05

No, the spinning of your space station does not warp space time. If that were the case then objects outside the space station should also be attracted to it (at least more than they are to a non rotating space station).

The effect that you see is purely due to the acceleration you experience in (nearly) flat spacetime due to the rotation. The equivalence principle implies that it will feel like a homogenous gravitational field.

• But Einstein said acceleration and gravity are equivalent. Gravity is a force that results from curvature of space time. Force also results from acceleration. If you state that gravity and acceleration are equivalent, doesn’t that imply acceleration warps space time? Commented Jan 10, 2019 at 3:31
• @Goldy That’s not what Einstein said and acceleration is most certainly not equivalent to gravity. Einstein said that gravitational mass and inertial mass are equivalent. This means that in small enough regions (where gravity can be assumed homogeneous) you will not be able to distinguish if the acceleration is die to a force or due to gravity. The standard test to find out which one is involved is to examine a big enough region of spacetime. Commented Jan 10, 2019 at 16:07
• I see. So the equivalence in that local region is what allowed Einstein to express accelerated motion as the same thing as an object at rest + a gravitational field? Commented Jan 11, 2019 at 19:50
• @Goldy yes. The biggest mystery that now remains is why this principle holds. But if they hold then we get some predictions --most of which have been confirmed. Commented Jan 12, 2019 at 5:28

Spinning an object does not warp spacetime.

In the case of our earth, spinning about the NS axis, we weigh slightly more (about 5.5 ounces) on the north and south poles as compared to the equator - because of centripetal acceleration. However, notice that spacetime around the earth is uniform, regardless of position. This is the best proof you've got.

No, in the examples that you are giving space-time is still approximately flat. The speeds and mass densities do not begin to approach the relativistic range.

The reason that the spinning spacecraft can produce accelerations comparable to the gravity of a mass like the earth is that the gravitational constant is so weak. At the earth's surface the acceleration due to gravity is only 9.8 m/s^2. However if you swing a ball on a 1 meter string at 1 revolution per second that acceleration is omega^2 * radius, or about 40 m/s^2 - over four times as much as gravity. This is why you can spin a jump rope or anything else without worrying about gravity once it gets going.

Well you have your own gravity (ever so little curved space) that you do not feel in a given uniform state. The rotating ship is continuously changing the state of your own space curve and you feel that against the surface. The resistance to change in state of your own gravity, manifests as inertia when subjected to a change in state.