# The upward force in General Relativity

#### Ascenxion

The upward force gives us the upward proper acceleration on the ground, making us non-inertial as Einstein predicted. This way, falling objects (inertial) seem accelerating. So, which one is correct?

The surface of the Earth is accelerating upwards, thus giving us that upward force. If this is true, the surface is accelerating at what rate?

OR

We are trying to follow the geodesics of spacetime, but the mechanical resistance of the Earth forbids that. Thus, we are pressed on the ground and there is an upward force exerted on us.

I would like an explanation on this.

Thanks.

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#### Fredrik

Staff Emeritus
Gold Member
Those two are just different ways of saying the same thing. (Both agree that falling objects are not accelerating). The acceleration of the surface of the Earth is the usual 9.8 m/s2 that you would use in calculations when you're using Newton's theory. (In Newton's theory, a falling object is accelerating at 9.8 m/s2 in the "down" direction. In GR, the surface of the Earth is accelerating 9.8 m/s2 in the "up" direction).

It's actually somewhat arbitrary which objects we should consider "accelerating" and "not accelerating". The definition of "not accelerating" used in GR is very natural because geodesics are the only paths with any kind of special significance in GR. (Just as "stationary in some inertial frame" defines a path through spacetime that has a special significance in pre-relativistic physics).

#### HallsofIvy

Einstein was careful to point out that this is a "local" equivalence. Because the equivalence is local, you cannot argue that two diametrically opposite points on the earth's surface are accelerating away from each other at 9.8 m/s2. Obviously we have been feeling a downward force for a long time. Just as obviously, the surface of the earth has NOT been accelerating upward during that time- the radius of the earth has not changed!

#### Ascenxion

So which one is more correct? Or are they both correct? I'm confused.

#### Fredrik

Staff Emeritus
Gold Member
General relativity is more correct than Newton's theory of gravity, and in GR it's more "natural" to say that the surface of the Earth is accelerating and that falling objects are not. But I still wouldn't say that the alternative is wrong. It's the natural choice in Newton's theory, which is a very good theory even though it isn't as good as GR.

I haven't thought hard about what happens when you choose a definition that isn't "natural" for the theory you're working with, but I think it just makes the mathematics more complicated when you try to define "force" by F=ma and use that concept in calculations. You probably won't be able to add forces the way you're used to, or something like that.

#### MeJennifer

you cannot argue that two diametrically opposite points on the earth's surface are accelerating away from each other at 9.8 m/s2.
Yes you can!

You are likely mistaken in thinking that acceleration implies a change in distance in time. That is not the case. For instance a rock at the end of a rope going in a circle keeps the same distance from the center but the rock is still accelerating.

A

#### Al68

The upward force gives us the upward proper acceleration on the ground, making us non-inertial as Einstein predicted. This way, falling objects (inertial) seem accelerating. So, which one is correct?

The surface of the Earth is accelerating upwards, thus giving us that upward force. If this is true, the surface is accelerating at what rate?

OR

We are trying to follow the geodesics of spacetime, but the mechanical resistance of the Earth forbids that. Thus, we are pressed on the ground and there is an upward force exerted on us.

I would like an explanation on this.

Thanks.
Both statements are correct, just like they would both be correct if they referred to the floor of an accelerating rocket.

#### Ascenxion

Thanks for the clarification, guys.

#### avillarreal

Just to be clear, if I have two bowling balls dropping, one in Colombia, one in Java (the exact opposite side of Earth), there is nothing bad about choosing a frame of reference on the first ball, but the only thing that happens is that my calculation of what happens with the ball in Java becomes a lot harder to calculate. The simplicity in the calculations on the Colombian side (a ball still in its frame of reference, with Earth accelerating towards it) is more than compensated by the difficulty of making calculations in Java (a ball accelerating at 2g's with no evident force on it).

It all becomes an exercise in futility, not a statement about the true frame of reference or the validity of General Relativity.

Correct?

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