Einstein's principle of equivalence; Standing on earth vs sitting in accelerated car

Question

When I am seated in a car that is accelerating in a particular direction I could, for example, throw a ball and it would appear to be flying the opposite way. With the windows covered etc. An observer on the earth's surface would tell me it's not the ball accelerating backwards, but the car speeding up forward.

As far as I understand, the principle of equivalence would also describe gravitational force in these terms(?). In other words, the reason for this 'pull' towards the centre of the earth is merely a fictitious force due to the earth not being an inertial frame of reference.

However, in a forward-accelerating car I find it easy to understand the ball's tendency to 'fly backwards' because of the direction of the car's acceleration. Two observers on the opposite poles of the earth still experience the pull towards the centre. The earth, as a non-inertial frame of reference can only be accelerating in one direction. Correct? Then why would people on two opposite sides of the earth still get pulled towards the centre?

Answer 1

I was confused by this too -- pop descriptions of the equivalence principle don't mention the problem where the gravitational field points in different directions in different places.

It is true that gravity is equivalent to acceleration, and that as a result, if you are freely falling, you feel like you're in an inertial frame. But this frame is only locally inertial; if the gravitational field varies, it breaks. That's okay, because the equivalence principle is a local statement: it says that gravity here is equivalent to uniform acceleration.

Answer 2

The Earth's gravitational field extends inward from all of space to the Earth's surface, with it's origin at the center of the Earth. The Earth's gravitational field is characteristic of space-time that has been "curved" by a massive object.

If you stand on the ground and let go of a ball, it falls away from you at the acceleration of 9.8 m/sec^2. Likewise, if you are accelerating in a spaceship at the rate of 9.8 m/sec^2, and you hold a ball out the window and let go of it, the ball will appear to fly backward at the rate of 9.8 m/sec^2. Gravitational mass is the same as inertial mass.

Inertial mass is defined by F = m * a.

Gravitational mass is defined by F = (G * m1 * m2) / r^2.

Both formulas yield the same "weight" or F in newtons. This is an illustration of the equivalence principle.

Answer 3

I think you are imagining the earth as one giant rigid inertial frame and that creates problems.

Let's look at the atmosphere, a giant doldrum over the pole to make it simple.

What keeps the air up there a certain height. Well there is a stronger pressure from the air below it than from the air above it.

Newtonian gravity would say the air stays at rest because the pressure gives a net force upwards that opposes the force of gravity on that parcel of air.

But GR instead says that there is a net force in the air so it accelerates upwards, which takes it to a region of spacetime where the metric looks the same and where there is exactly the same pressure imbalance.

This is a completely different story, but much more accurate. If there is mire pressure from below than from above and no other forces then you go up.

And that happens around the south pole and the north pole each parcel of air actually accelerates up because each parcel of air has more pressure from below than from above.

And that's why you don't fall to the center if the earth, the ground below you is in a compressed state and exerts pressure on your shoes which themselves are compressed and exert pressure on your feet and your keg is compressed and exerts a pressure in you midsection and so on. You are accelerating upwards in an inertial frame. Its just that the things around you are accelerating upwards to.

From your perspective the inertial frame is accelerating downwards. But the earth (the ground) is not moving inertially, the top layer of the earth is also feeling more pressure from below it than it feels from above and the rocks and such below you are also accelerating upwards.

So the problem is that you simply didn't let go of the idea of the earth as a rigid object that determines a frame. Each little infinitesimal patch of spacetime is an inertial frame and to you they look like they are accelerating downwards, unless you happen to be moving inertially. Which is uncommon.

Instead of infinitesimal inertial frames you can work with a finite coordinate neighborhood but then you need to use the curved geometry of GR.

So people are not being pulled to the center of the earth. The people and the parts of the earth are generally being accelerated (acceleration as always relative to an inertial frame) outwards because of things like pressure gradients. When they don't feel pressure gradients and such, they move inertially which might loom like acceleration to people that aren't moving inertially.

And if people standing on the earth look like they care accelerating in different directions on opposite sides of the earth it is just because each is experiencing pressure gradients pushing them away from the center because that's how the matter around them is arranged.