# Twin Paradox With A Beacon

1. Dec 19, 2012

So once again, I'm trying to wrap my head around the asymmetry in the twin paradox problem. Here's my setup. The two twins start on a very massive but small planet such that they always have an unobstructed view of each other.

1. They each attach a beacon to themselves that will briefly flash every month based on their individual clocks.

2. One twin goes in orbit around the planet at a distance to obtain a circular orbit with a period of 1 month and a velocity with a Lorentz factor of 30.

[Given the constraint of a circular orbit with the specified period and Lorentz factor (and neglecting GR), this places the orbit at 1.24e14 meters about a planet with mass 1.67e41 kg].

3. Neglecting the 12 hour delay that is approximately takes light to travel the distance between the twins, the twin on the planet will see the twin in orbit's flash in the same place in the sky, but by the stationary twins clock, this will only happen every 30 months.

4.Once again, neglecting the same 12 hour delay for light to make its journey, the twin in the ship will see the stationary twin flash his beacon 30 times for every orbit he makes.

Now the question - if we treat the orbiting twin as the "stationary" observer, then shouldn't he observe the other twin's beacon (originally the twin on the planet) to be flashing very slowly? The best explanation someone has given me in the past is that because one of the twins is accelerated, there is an asymmetry. But then, that suggests there's some preferred reference frame relative to accelerated observer.

The only argument I can think of to support some type of preferred frame would be if I had super strength and pushed off against the earth with enough force to accelerate myself beyond the planet's escape velocity.

In the planet's frame, the kinetic energy of this two body system is 0.5* my mass * v^2
(assuming I'm not moving at speeds near c)

In my frame, the kinetic energy of the two body system is 0.5 * Earth's mass * v^2

Clearly the product of the force I applied over with whatever distance that I applied it over isn't equal to the energy in the latter of the two frames.

2. Dec 19, 2012

### pervect

Staff Emeritus
You can't realistically just "ignore GR" here, unless you take the step of having one twin in a powered orbit - i.e. replace gravity with thrust.

If you do that, then I hope it's obvious that the twin in the powered orbit is accelerating....

Have you read one of the zillion posts on the twin paradox, specifically about the relativity of simultaneity?

The point you should be getting is that in SR, the twins notion of "now" is different - and this allows both twins to fairly claim that the other twin's clock is running slow.

3. Dec 20, 2012

### ghwellsjr

I haven't checked your arithmetic so I'll just assume it's correct but in this last point the twin in the ship will see the stationary twin flash his beacon 1 time for every orbit he makes (not 30 times).

You did say the orbital period was 1 month which I assume you meant in the planet's frame and you said the twin on the planet flashes his light every month so there's a one-to-one correspondence between each orbit and each flash coming from the planet. However, the orbiting twin will think each orbit takes 1/30 of a month (which is a day?) so he will see the stationary twin flash his beacon 30 times for every one of his months.
There is an asymmetry but why do you think that suggests there's some preferred reference frame relative to the accelerated observer? The planet's rest frame is the easiest to analyze the scenario in but that doesn't make it preferred. You could still transform the scenario to another reference frame moving with respect to the planet.
I don't understand what you are trying to do or why you think there is a problem. Just remember, the planet twin is inertial while the orbital twin is not and this has nothing to do with any preferred frame.

4. Dec 22, 2012

### Cleonis

Your question is: what factor introduces the asymmetry?

As we all know Special Relativity asserts that a concept of velocity relative to spacetime does not enter the theory as a matter of principle.

However, this does not extend to the following two concepts:
- Difference in pathlength travelled
- Acceleration relative to spacetime

It's just as fundamental to SR that the above two do enter the theory.

For the clock in orbit a smaller amount of proper time elapses, correlating with the orbiting clock covering a larger spatial distance. Of course, there is no concept of position relative to spacetime, but when you have two objects you can always ascertain if one of them is covering a larger spatial distance than the other.

Difference in pathlength travelled and acceleration relative to spacetime are correlated; you cannot have one without the other.

As a heuristic I present a model that features the above described properties self-consistently. I'm not saying the model is what is happening physically, I'm just saying the model covers the assertions and that it's self-consistent.

Consider the following infinite set: all coordinate systems of a Minkowski spacetime. Each member of that set can be transformed to another member of that set with a Lorentz transformation. This set is often referred to as 'the equivalence class of inertial coordinate systems'.

All coordinate systems of this equivalence class have a velocity relative to each other, so there is no concept of some veloicity relative to the equivalence class as a whole.

However, this equivalence class as a whole is an unambiguous reference for acceleration. While there is no single preferred inertial coordinate system, there is a single reference for acceleration.

It's natural to gravitate to the assumption that if velocity relative to spacetime does not enter the theory, then acceleration relative to spacetime can't be in the theory either. That's the assumption that needs to be dropped: acceleration relative to spacetime does enter.

Once again the orbiting clock (being compared with a clock on a straight worldline): the difference in the amount of proper time that elapses is given by the Minkowski metric. The asymmetry correlates with difference in travelled pathlength.

Cleonis