# Free Fall vs accelerated reference frame question.

#### BTBlueSkies

I had a though last night that kept me awake trying to figure out what I was trying to understand. I have been out of school many years now so I have forgotten much.. Here is my question.

Lets imagine there is a 30 meter diameter hole (complete with a vacuum) in my front yard that goes through the center of the earth all the way through to the other side. Basically, a hole through the center of the earth. This earth is not rotating around our sun, it is isolated in some space all by its self. Nor is it rotating etc.

Lets imagine we have three persons.. and I am on of the three.

Lets imagine there are two hollow 2 meter metal spheres in my front yard. One is sitting on the ground next to the hole, and the other is suspended over the hole at surface level by an electromagnet.

Lets imagine that in each of the two spheres contains one of the people, and I am standing in the open air in my front yard watching these two spheres and such. We all have an accelerometer and a notepad and a perfect clock.

One last detail, once inside the sphere, the person has no knowledge of what the situation is. It's like they woke up magically in a sphere not knowing if they were on a planet, in this universe, etc. They simply have no idea what is up except they are in a round spherical container, and they are going to try to figure out what is up.

I think that sets up for my question

So, I have a button to turn off the electromagnet which wakes the two people up in the spheres at the same moment, and the clock starts a timer.

So I was imagining the situation for the two people in the spheres.

Sphere A was over the hole and when the magnet was released he awakes to find him self in a gravity free sphere where he is thinking he is perfectly still floating somewhere in space. His accelerometer is reading zero on x,y and z.

Sphere B was on the ground and thus wakes up to find him self in a sphere sitting on one side. He is a space man so he thinks he is in a ship taking off, and doesn't realize he is just sitting on the ground. So he does the math with his clock and the accelerometer and decides is traveling ever faster in the direction opposite where he is sitting.

The interesting thing to me in this question is that I, the third guy, knows exactly what the situation is as I am watching it. When the falling sphere passes through the earth to the other side, it will just come back to where it started and I will catch it with the electromagnet and get the guys out of the spheres and then we will compare notes as to what we experienced.

Lets imagine it took 90 minutes for the falling sphere to return to the electromagnet to be caught, the rough time it takes a satellite to orbit the earth per say, the exact number is not important I don't think.

In our discussion after, over a beer, we relate our experiences.

The guy in Sphere A just figured he was stationary in space while he was actually in free fall motion unpretrubed on his trip. So he just noted the 90 minutes of floating perfectly still. I noted that he traveled to the other side of the world and back.

The guy in Sphere B just figured he was in a rocket and was accelerating in a straight line at 1g for the 90 minutes and figured he had traveled about 142,884km and should have been going about 190,512kph at about that time. He also calculated that since he weighed exactly 1_kg (skinny dude), that it took about 1.4 x 10^9 Joules of energy to get him going that fast. I noticed he didn't change location or speed, and no energy change occurred.

So here is my question .....

I understand the guy in Sphere A could also have guessed he was in a sphere in free fall and that the guy in Sphere B could have guessed he was just sitting on a massive body with a gravity of 1g. This is the equivalency principle I believe.

What I don't get is the energy change part of the Sphere B situation. Sphere A did have a kinetic energy change during his trip however he returned to the original energy and thus no net change occurred. Sphere B either 'might have' or 'might not have' had an energy change. I am not sure how to reconcile this idea.

Additionally, I am thinking the Sphere A clock might have slowed down during the 90 minutes, as he was in motion. However, the clock in Sphere B could have slowed down also, or maybe not, as he may have been in motion or may not have been in motion.

I guess, what I am puzzled with is that we have lived in an accelerated field all of our lives here on Earth, and if accelerated fields are the same whether gravitational or inertial, and if I lived my whole life in that Sphere B, could I be going close to the speed of light in the space inside the sphere as I have had the idea that I was accelerating all these years at 1g.

As you can probably see, I am a bit confused, but interested :)

Thanks for considering my question.

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#### mfb

Mentor
The guy in Sphere B just figured he was in a rocket and was accelerating in a straight line at 1g for the 90 minutes and figured he had traveled about 142,884km and should have been going about 190,512kph at about that time. He also calculated that since he weighed exactly 1_kg (skinny dude), that it took about 1.4 x 10^9 Joules of energy to get him going that fast.
As seen by the frame of "B at the end of this hypothetical trip", he startet with 1.4GJ and lost this energy during the trip.
As seen by a different observer, he startet with 1/4 this value, lost it during the first half of the trip, and got the same energy again in the second half.
Kinetic energy and differences in it depend on the reference frame.
And what really happened was no energy transfer at all of course - it does not need energy to let a sphere rest on the ground. The situation looks the same from the inside, but with outside knowledge (= where energy would be exchanged) you see a difference between the accelerated trip in space and resting on the ground.

I understand the guy in Sphere A could also have guessed he was in a sphere in free fall
Being in space far away from larger masses is just a special case of free fall, so A knows he was in free fall all the time.
Additionally, I am thinking the Sphere A clock might have slowed down during the 90 minutes, as he was in motion.
He was also lower in Earth's gravity well.
However, the clock in Sphere B could have slowed down also, or maybe not, as he may have been in motion or may not have been in motion.
He has not been in motion relative to us. No slowdown.

and if I lived my whole life in that Sphere B, could I be going close to the speed of light in the space inside the sphere as I have had the idea that I was accelerating all these years at 1g.
Right. Luckily we don't live in a sphere.

It is possible to distinguish between the setups using the finite size of the sphere: both will see tidal gravity which allows to see that they are close to some massive object.

#### DrGreg

Gold Member
As you can probably see, I am a bit confused, but interested :)
The solution to your confusion is that energy is relative to the observer measuring it (or, more precisely, the coordinate system being used).

This is true even in Newtonian mechanics (Galilean relativity) where kinetic energy depends on velocity, but velocity depends on the observer. If an object changes speed, one observer might calculate an increase in kinetic energy while another observer might calculate a decrease. Within each coordinate system, energy is conserved but you'll get different values for the total energy in each system.

That's for inertial (non-accelerating) observers. For accelerating observers you also need potential energy to get the equations to balance.

#### Dale

Mentor
Hi BTBlueSkies welcome to PF,

I think there are two concepts that might be tripping you up. The first is that energy is not frame invariant. So if you get that something has 1.4e9 J of KE in one frame and 0 J in another, that is fine.

The second is that the equivalence principle only applies locally in spacetime. That means over distances and times where the curvature of spacetime is negligible. Over several years the curvature is clearly non negligible, so we don't expect to see the universe whipping past us at nearly c.

#### BTBlueSkies

It is possible to distinguish between the setups using the finite size of the sphere: both will see tidal gravity which allows to see that they are close to some massive object.
I can see this. For example, if the sphere was full of an array of accelerometers, say 1cm apart in x,y,z filling the sphere, under propulsion acceleration they would all read the identical values. However if I was near a massive object, and I subtracted the least common vector from all the accelerometer readings, I would see the tidal gravitational gradient and thus know I was in the gravitational field of another entity?

I am not sure about the 'least common vector' comment, but the idea I am thinking is that if there was a combination of propulsion acceleration and gravitational acceleration, then if I could identify the 'propulsion' part, and then just the gravitational components existed then there would be common part to the gravitational field as well that could be subtracted leaving a final 'tidal gradient'.. or so. This part is not so critical to my understanding though.

I am thinking what is critical, is that if the 'gravitational tidal' gradient field can be 'always' identified, then does that invalidate the equivalence principal as you can always know which is the situation thus they are not identical?

I understand idealized equivalence, but idealized has limits I am suspecting.

I am also assuming we have infinite precision instruments and such.. so scale can be removed from the list of 'what ifs' :)

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#### BTBlueSkies

That's for inertial (non-accelerating) observers. For accelerating observers you also need potential energy to get the equations to balance
I meant to add this idea as I knew it would be mentioned. I do understand the energy vs observer component.

The guy in the yard can know the real situation, but the guy in the sphere cannot know. He cannot know if he is moving or stationary on a massive body.

However, the gravitational tidal idea leads me to believe he 'can with certainty' know.

I am wondering if my confusion is knowing what happens to the gravitational energy imparted on the guy that is in the sphere on the ground. If he 'could not know' if he was accelerating or on a massive body, that energy of acceleration is either going into his 'kinetic and potential' energy or it is .. hmm.. being absorbed by the material between the center of mass and the sphere? Maybe it's doesn't actually exist, the energy idea I have from the gravitational acceleration is maybe not the right idea?

Thanks for the replies. :)

#### mfb

Mentor
I'm not sure if you can get all of this with multiple accelerometers. There could be some ambiguity left if you allow gravity plus rockets at the same time.
I am thinking what is critical, is that if the 'gravitational tidal' gradient field can be 'always' identified, then does that invalidate the equivalence principal as you can always know which is the situation thus they are not identical?
The equivalence principle is local - using multiple accelerometers at different places is not local any more.
the energy idea I have from the gravitational acceleration is maybe not the right idea?
I'm not sure what you mean with "energy idea", but it does not sound right.

#### BTBlueSkies

I'm not sure if you can get all of this. There could be some ambiguity left if you allow gravity plus rockets at the same time.
Ya, that adds a wrinkle. Say remove the propulsion component, would a gravitational tidal gradient always be present? I am thinking yes as while we can construct an idealized local uniform gravitational field, with infinite precision instruments, we would always find the gradient at some x number of decimal places.

If this is absolutely wrong, that would be key for me to know though.

Thanks.

#### BTBlueSkies

The equivalence principle is local - using multiple accelerometers at different places is not local any more.
ah... so even two accelerometers 1cm apart violates local? I get 'local' now.

Thanks!

#### mfb

Mentor
Ya, that adds a wrinkle. Say remove the propulsion component, would a gravitational tidal gradient always be present? I am thinking yes as while we can construct an idealized local uniform gravitational field, with infinite precision instruments, we would always find the gradient at some x number of decimal places.
Sure.
so even two accelerometers 1cm apart violates local?
Right.

#### BTBlueSkies

Ya, that helps. I was reading on wiki the following.. http://en.wikipedia.org/wiki/Equivalence_principle

The Einstein equivalence principle
What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:[32]

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
Here "local" has a very special meaning: not only must the experiment not look outside the laboratory, but it must also be small compared to variations in the gravitational field, tidal forces, so that the entire laboratory is freely falling. It also implies the absence of interactions with "external" fields other than the gravitational field.

The use of the word 'laboratory' and 'local' was making me think local was less specific than a point source.

I'll keep at it..

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