# Twin paradox caused by gravitational difference in space

I am not a physicist, but have always been curious about the twin paradox. So, here is my question. There are two twins in space - Twin A and Twin B (both stationary). They are apart from each other but close enough to see each other.

At first, they experience the same gravity. But, suppose that the place where Twin B is standing suddenly experiences a huge localised increase in gravity for some reason (e.g. a tiny black hole suddenly appears or passes by near Twin B), and after a few minutes, the gravity Twin B experiences will go back to the same gravity as Twin A. After that event, they can see that Twin A is aged faster than Twin B (also during that event, they see that Twin A is aging faster/twin B is aging slower).

If this is the case, then can we say that "acceleration" or changed frames of reference are not required to resolve the twin paradox? It is not about who is "accelerating" superficially, and they can be viewed as they are both in inertial frames of reference from each other's perspective... the asymmetry situation is therefore due to the changing curvature (landscape) of the space-time, not due to the "acceleration" nor due to the changed frames of reference by Twin B...

If this is so, then can we argue that this also applies to the standard story of the Twin Paradox where Twin A stays on the earth and Twin B goes off the space and come back?

Kind regards

Here is my follow up thought after some inputs from other people in this forum. I came to this conclusion...but let me know if I am mistaken.

I really appreciate your feedback on this to deepen my understanding on this topic.

Thank you!

• So, twin B remains stationary relative to twin A while the gravitational field moves by? If so, then twin B is no longer in a freely falling reference frame, and very much is accelerating relative to the background geometry. Jul 8 '15 at 22:19
• You always have to be very careful about what you're assuming in questions like this. What does it mean to say that "after a few minutes", the gravity Twin B experiences goes back to the same gravity as Twin A? A few minutes according to whom? Jul 8 '15 at 23:54
• Hi, WillO Say, a few minutes according to Twin A. Either way, in the end, they will see that their clocks do not match. Jul 10 '15 at 13:47

So, just to recap the Twin Paradox, it is a variation of the paradoxes of relative motion of reference frames Alice and Bob, created by the statement "Alice sees Bob's clocks moving slowly, but Bob also sees Alice's clocks moving slowly." The simplest such paradox, in my opinion, is "what if Alice calls Bob up and they talk on the phone? One of them surely notices the other talking in slow motion, which is it?" The resolution hinges on the fact that the phone only uses waves which propagate at the speed of light or slower, so their communication with each other is first and foremost relativistic-doppler-shifted, so they both see communication delays which dynamically mask their ability to detect which of them is "right".

In the Twin Paradox we say, "aha, but now after a while we bring Alice and Bob back together: surely one of them thinks the other one is older than they "should be", but there is still no reason to prefer one of their stories to the other one."

Your variant of the Twin Paradox is therefore talking about something rather different: in such a case, both Alice and Bob can determine that Bob's clock is moving slower than Alice's due to a nearby black hole. It is therefore also not of very much help in resolving the actual Twin Paradox, which does get resolved by either having one of them pass through a periodic boundary condition in spacetime or else by having them each accelerate in different ways, which de-synchronizes remote clocks as they used to understand them.

If this is the case, then can we say that "acceleration" or changed frames of reference are not required to resolve the twin paradox?

and you say:

they can be viewed as they are both in inertial frames of reference from each other's perspective

The answer is that acceleration is required to resolve the twin paradox and the two observers are not both in inertial frames.

It is a fundamental principle of general relativity that acceleration and gravity cannot be distinguished. This is called the equivalence principle. For convenience we'll assume that $A$ is far enough from the mini black hole that they can be regarded as in an inertial frame. Then we have only to ask about what happens to $B$.

Suppose $B$ is in a sealed spaceship so they cannot see anything outside. Suppose now $B$ lets go of a ball that they are holding. What happens to the ball?

1. if $B$ is in an inertial frame the ball stays where $B$ released it i.e. the ball doesn't move relative to $B$.

2. if $B$'s spaceship is accelerating because the rocket motor is firing then the ball drops to the floor

3. if $B$ is hovering at some distance from a black hole then the ball also drops to the floor

The point is that $B$ cannot tell the difference between (2) and (3), that is $B$ cannot tell whether they are being accelerated by a rocket motor or are feeling a gravitational field. This is fundamental to general relativity and is part of the equivalence principe I mentioned above.

So $B$ is not in an inertial frame, even though they are not moving relative to $A$. $B$ is in an accelerated frame and the time dilation is related to the acceleration.

• Hi, John and others. Jul 10 '15 at 13:52
• @Abacus: just saying hello? Or did you have something further you wanted to ask? Jul 10 '15 at 15:14
• Hi, John and others. Thank you for your replies. Ok, so, according to the equivalence principle, acceleration and gravity are indistinguishable because both bend/create curvatures of space-time. And, if my understanding is correct, visually observable "acceleration" (or do we have any other term for it?) is not the same as the acceleration when we talk about the theories of relativity and equivalence principle, correct? and also frames of reference are different from perspectives/point of view, correct? Jul 10 '15 at 21:42
• If Twin B is in a rocket flying away from the earth where Twin A is staying. If there is a window in Tiwn B's rocket, B can surely observe that Earth (Twin A) is flying ("accelerating") away from Twin B who can consider himself as stationary (except for the g force he experiences) and his perspective/point of view remains the same. However, the frame of reference changes for Twin B due to the acceleration/gravity... Is my understanding correct? Jul 10 '15 at 21:43
• @Abacus: acceleration and gravity are locally equivalent because locally spacetime is approximately flat. As soon as you look farther away than your immediate vicinity the difference can be measured. Jul 11 '15 at 6:07