Gravitating Body Acceleration: Bob vs Jim

[jartsa]

Hey guys, what do you think about this:

Let's put a planet in empty space, on the surface of the planet we put a a guy named Bob, Bob's task is to accelerate the planet by pulling on a provided rope. He must extend his arm, grab the rope and bend his arm at maximum force.

Bob's identical twin Jim is sitting at the center of the planet pondering: If I, while in this low position, was doing the same task as Bob, it would take more time for me to bend my arm, which would result in lower velocity of the planet. The velocity change would be smaller by time dilation factor, and the time during which the velocity change occurs would be longer, so acceleration of the planet would be smaller by time dilation factor squared ... and observers anywhere would agree about the ratio of the accelerations caused by me and Bob, so it must be so that my muscles can produce a force: Bob′s force ∗ square of time dilation factorOne addition: A giant measuring stick is hanging near the planet, giving Bob and Jim a good idea of the motion of the planet.

 

[Nugatory]

Hey guys, what do you think about this:

Let's put a planet in empty space, on the surface of the planet we put a a guy named Bob, Bob's task is to accelerate the planet by pulling on a provided rope. He must extend his arm, grab the rope and bend his arm at maximum force.

Bob's identical twin Jim is sitting at the center of the planet pondering: If I, while in this low position, was doing the same task as Bob, it would take more time for me to bend my arm, which would result in lower velocity of the planet. The velocity change would be smaller by time dilation factor, and the time during which the velocity change occurs would be longer, so acceleration of the planet would be smaller by time dilation factor squared ... and observers anywhere would agree about the ratio of the accelerations caused by me and Bob, so it must be so that my muscles can produce a force: Bob′s force ∗ square of time dilation factor

More time, smaller, lower, longer, acceleration... all nice words, all completely meaningless until you've said what they're relative to. So I'll ask: Relative to what?

There's only one meaningful definition of the strength of someone's muscles, and that's the proper acceleration that they can impart to a mass otherwise in free fall (this definition is equivalent to the amount by which they can deflect a spring of known spring constant, so covers just about all the other ways that we measure strength). It's the same for Bob and Jim under the conditions you've specified here.

 

[jartsa]

More time, smaller, lower, longer, acceleration... all nice words, all completely meaningless until you've said what they're relative to. So I'll ask: Relative to what?

Jim sees Bob to make a fast arm motion, which hurls the planet at fast velocity, fast according to Jim. When Jim tries to do the same thing, the velocity in this case according to Jim is lower than in the first case according to Jim.

There's only one meaningful definition of the strength of someone's muscles, and that's the proper acceleration that they can impart to a mass otherwise in free fall (this definition is equivalent to the amount by which they can deflect a spring of known spring constant, so covers just about all the other ways that we measure strength). It's the same for Bob and Jim under the conditions you've specified here.

The physical ability of a planet dweller to move his planet (by pulling some suitable ropes) in order to dodge asteroids. I think that is meaningful. How does that depend on gravitational time dilation?

 

[jartsa]

I received complaints about the thought experiment - only proper times are meaningful, or something like that.

So how about if Bob and Jim know that their proper mass is 70 kg.

They figure out the mass of the planet by:

1: Bob moves to the left on the surface of the planet and an assistant measures how much the planet moves to the right, by observing distant stars.

2: Jim moves to the left at the center of the planet and an assistant measures how much the planet moves to the right, by observing distant stars.Now I think the measured masses will differ by the relative time dilation factor between Bob and Jim raised to the second power. And the larger mass measured by Jim compared to Bob explains why it's more difficult for Jim to accelerate the planet compared to Bob.

 

[PeterDonis]

They figure out the mass of the planet by:

1: Bob moves to the left on the surface of the planet and an assistant measures how much the planet moves to the right, by observing distant stars.

2: Jim moves to the left at the center of the planet and an assistant measures how much the planet moves to the right, by observing distant stars.

How far do Bob and Jim move? How is this distance measured?

 

[jartsa]

How far do Bob and Jim move? How is this distance measured?

Bob moves some proper distance, let's say 10 m, measured by Bob, using a measuring tape.
Jim moves some proper distance, let's say 10 m, measured by Jim, using a measuring tape.To simplify this thing: An inertial navigation system placed near Bob measures the proper distance that the planet moves when Bob moves a proper distance 10 m.
PlanetMassAccordingToBob / BobMassAccordingToBob = BobDisplacementAccordingToBob / PlanetDisplacementAccordingToBob

 

[jartsa]

I guess you guys do not appreciate my reasoning in post #1. Well I want to point out that Jim, the guy at the center of the planet, must do lifting work as he accelerates the planet, as the acceleration work is done to a thing that is above him, as the whole planet is above him.

Energy is lifted from Jim to Planet by Jim.

So this is supposed to convince everybody that the acceleration are different when the pulling is performed at different altitudes.

 

[Nugatory]

I guess you guys do not appreciate my reasoning in post #1. Well I want to point out that Jim, the guy at the center of the planet, must do lifting work as he accelerates the planet, as the acceleration work is done to a thing that is above him, as the whole planet is above him.

Above him in what sense? "Above" means "the opposite of the direction that gravity is pulling", and because of the shell theorem there's no gravitational force for Jim to resist. This entire thought experiment comes down to having two identical mechanisms (Jim's muscles and Bob's muscles) exerting force on an object of given mass while at different gravitational potentials (Jim is deeper in the potential well).

So this is supposed to convince everybody that the acceleration are different when the pulling is performed at different altitudes.

The coordinate accelerations are different. The proper accelerations are not different, so the conclusion you came to in the original post (identical muscle produces different force) does not follow.

 

[Nugatory]

One addition: A giant measuring stick is hanging near the planet, giving Bob and Jim a good idea of the motion of the planet.

It will tell them whether the planet is moving relative to the measuring stick, but unless you also provide a clock, it won't tell them how rapidly the planet is moving and/or accelerating. Then you have to decide how the twins are moving relative to the clock (neither can be at rest in an inertial frame in which the clock is at rest as both are experiencing proper acceleration while they're pushing on the mass) and agree on a simultaneity convention.

 

[jartsa]

Above him in what sense? "Above" means "the opposite of the direction that gravity is pulling", and because of the shell theorem there's no gravitational force for Jim to resist. This entire thought experiment comes down to having two identical mechanisms (Jim's muscles and Bob's muscles) exerting force on an object of given mass while at different gravitational potentials (Jim is deeper in the potential well).

For fun, let's consider a case where the planet is below instead of above .

Let's say there are two planets in empty space, and Bob and Jim standing on the surfaces of the planets, and a rope between Bob and Jim.

And then we have Alice standing at the middle point of the planets observing what happens as Bob and Jim start pulling on the rope.

What happens according to Alice is:

There is a Bob-Jim-system that is doing work on Planet-Planet-system. Bob-Jim-system loses rest-mass, Planet-Planet-system gains rest-mass.

Rest-mass kind of moves from Bob-Jim-system to Planet-Planet-System. Planet-Planet-system is below Bob-Jim-System, at lower gravitational potential, because its own gravity.

The lost potential energy of the rest-mass that moved form Bob-Jim-system to Planet-Planet-system must become some other form of energy. It becomes kinetic energy of the planets.
So that was the idea.

 

[Nugatory]

For fun, let's consider a case where the planet is below instead of above .

Now you're describing a completely different situation, one that is symmetrical and in which gravitational time dilation between the twins is irrelevant.

You will be able to make sense of this one if you remember that:
1) Kinetic energy is frame-dependent.
2) Rest mass is not frame dependent. It is defined to be the mass of an object as measured in a frame in which its momentum is zero. A corollary is that when you're talking about rest mass "moving from the Bob-Jim system" to the "planet-planet system", you'll just confuse yourself.
3) Introducing Sue into the picture just gives yoiu yet another coordinate acceleration that you can use to confuse yourself,

 

[jartsa]

Now you're describing a completely different situation, one that is symmetrical and in which gravitational time dilation between the twins is irrelevant.

You will be able to make sense of this one if you remember that:
1) Kinetic energy is frame-dependent.
2) Rest mass is not frame dependent. It is defined to be the mass of an object as measured in a frame in which its momentum is zero. A corollary is that when you're talking about rest mass "moving from the Bob-Jim system" to the "planet-planet system", you'll just confuse yourself.
3) Introducing Sue into the picture just gives yoiu yet another coordinate acceleration that you can use to confuse yourself,

The important thing in the scenario is that Bob-Jim-system and Planet-Planet-system stay at rest, because the two planets have the same mass. I forgot to say that. The center of masses of the systems stay at rest.

It's a fact that rest-mass of Bob-Jim-system is smaller at the end than in the beginning, and rest-mass of Planet-Planet-system is larger at the end than in the beginning. (Oh yes, I should have said that Bob and Jim are not counted into Planet-Planet-system)

What happens between beginning and end? Any ideas?

 

[PeterDonis]

Let's say there are two planets in empty space, and Bob and Jim standing on the surfaces of the planets, and a rope between Bob and Jim.

If the planets start out at rest relative to each other, they will move towards each other even if Bob and Jim do nothing.

Planet-Planet-system is below Bob-Jim-System, at lower gravitational potential, because its own gravity.

The concept of "gravitational potential" is not well-defined in this system, because it's not static (see above). However, if you view it all as a single system (which includes both planets and Bob and Jim) from someplace very far away, the total energy of the system as a whole is constant.

The important thing in the scenario is that Bob-Jim-system and Planet-Planet-system stay at rest, because the two planets have the same mass. I forgot to say that. The center of masses of the systems stay at rest.

No, they won't. The planets will fall towards each other because of their gravity.