# Is this an inertial frame of reference in relativistic context?

What I've learned from our special relativity lectures is that an inertial frame of reference in relativity is one that experiences no gravitational forces. Also, it is a frame where if a particle is initially at rest, it will remain at rest. And if it is initially in motion, it will remain in motion with constant speed and same direction.

My question is if my understanding is right about the following:

1. An elevator in free fall is NOT an inertial frame of reference because it experiences gravitational forces
2. A rocket that accelerates upward with acceleration g = 9.81m/s IS an inertial frame of reference since the earth pulls it down with the same acceleration so this cancels and there's like zero net acceleration so it doesn't experience any gravitational forces.

I'm sorry if my trains of thought sound stupid. I'm just really struggling with relativity.

• Jul 1, 2015 at 12:43
• Jul 1, 2015 at 12:47

Special relativity and general relativity have different views about inertial frames, but in some ways the general relativity take on them is (perhaps surprisingly) easier to explain. So I'll start with GR then extend the description to SR.

In general relativity there are usually no global inertial frames i.e. it is impossible to construct a frame that looks inertial everywhere. However it is always possible to construct a locally inertial frame. This is a frame that looks inertial as long as you consider only the spacetime in your immediate vicinity. To check if your frame is locally inertial you surround yourself with a sphere of test particles then watch to see what happens to them.

If the sphere of test particles remains unchanged in relation to you then your frame is locally inertial.

However if the sphere of particles moves with respect to you then your frame is non-inertial because it's accelerating. Finally if the sphere of particles changes shape then the frame is non-inertial because there are tidal forces acting.

Now see how this definition applies to your two specific questions:

1. if you and the test particles are falling down an elevator shaft then you and the particles both accelerate with the same acceleration of $g$. That means your frame is locally inertial. Exactly this reasoning applies to an astronaut aboard the International Space Station. Suppose you're in a part of the the ISS where there are no windows. If you throw a ball it's going to travel in a straight line at constant speed, even though the ISS is moving in a circular orbit around the Earth. If you don't look out of the windows you couldn't tell you weren't floating freely in space far from any masses.

2. If you're in a rocket hovering at a fixed distance above the Earth then when you let go of the test particles they'll fall to the floor. This is an accelerated frame not a non-inertial frame. If you throw a ball it won't travel in a straight line at constant speed.

So actually the answers to your questions are the other way around to what you thought.

Incidentally, if you look more closely at the falling elevator frame in your question (1) you'll realise it isn't inertial either. The acceleration due to gravity changes with distance from the centre of the Earth, so the test masses nearer the centre will accelerate slightly faster while the ones farther from the centre of the Earth will accelerate slightly slower. The result is that your sphere of test particles changes shape and gets stretched out into an oval. This is an example of tidal forces.

However if the size of your test sphere is small the tidal forces will be small. If we make the radius of the sphere really tiny the tidal forces will be undetectably small. This is what we mean by locally inertial - the frame looks inertial if we consider a small enough region of it.

Now let's consider special relativity. In this case there is no gravity so if you were in the elevator shaft you'd just float there. In this case your sphere of test particles would not move with respect to you no matter how big you made the sphere. So this frame is inertial, but unlike the GR case it's globally inertial. Since there's no gravity you could make your test sphere light years in size and it still wouldn't change with time.

The absence of gravity affects the rocket as well. With no gravity the rocket wouldn't hover above the Earth but instead it would go shooting off into outer space at an acceleration of 9.81 m/s$^2$. However inside the rocket you couldn't tell the difference. When you released your test articles they'd still fall to the floor in the same way, so your frame is still non-inertial.

Actually the fact that you couldn't tell the difference between gravity and the rocket accelerating away is a key part of general relativity, and it's called the equivalence principle.

In Newtonian physics, neither are inertial frames. In relativity, only the first one is an inertial frame.

Edit:

Let me clarify. An inertial reference frame is a frame that "follows" an uniform motion, i.e. a motion where there could be said not be affected by any net force. In Newtonian physics gravity is a force and objects in free fall are thus considered to accelerate; they are affected by a net force. In relativity, gravity is not considered a force and orbits are merely straight geodesics in curved spacetime.

In the situation of the accelerating rocket, it can be said to accelerate (and be affected by a net force) in both Newtonian physics and relativity. It isn't a inertial frame of reference because of this.

• Dear Madde, this answer made its way into the low-quality review queue, because sans an explanation, this isn't very useful. Jul 1, 2015 at 15:52
• @TheDarkSide I have updated by answer and clarified by position. Jul 2, 2015 at 0:18