August 25, 2008

Special Relativity and the Twin Paradox

Dear Dr. Craig,

I am reading your wonderful book, Time and the Metaphysics of Relativity (Kluwer Academic Publishers, 2001), and I have a question about the Twin Paradox and the Paradox of the Three Brothers. (The Twin Paradox posits that a twin who travels into space for a protracted period at near light speed, then returns to earth, will be much younger than the twin who remained earth-bound.)

In order to illustrate the paradox, you also describe The Paradox of the Three Brothers (page 55, figure 3.4), which I take to be merely a variation of the Twin Paradox, intended to show that the paradox persists even if acceleration is eliminated from the thought experiment.

In the Paradox of the Three Brothers, A is on Earth and thus "at rest." B passes A at a high uniform rate of speed at E1, and at E1, A's and B's "clocks agree." B then proceeds away from Earth to E2, where he passes C who is on his way back to Earth, and at E2 B's and C's "clocks agree." C continues back to Earth, which he reaches at E3, and the question is whether A's and C's "clocks agree" at E3, and we are supposed to understand that they do not agree.

I see several problems with the Paradox of the Three Brothers. Can you straighten me out?

So hypothetically, A passes B at a uniform rate, at which time their clocks agree. But how did B get into space in order to return to Earth to pass A at a uniform rate? They are brothers, remember, and therefore B must have started out on Earth. But if B started out on Earth, then how could their clocks agree? Doesn't the Twin Paradox (which the Paradox of the Three Brothers is ostensibly intended to illustrate) show that B is younger than A at E1? What, then, do we mean when we say that their "clocks agree"?

If (as stated in the hypothetical) A, B, and C are brothers, then B and C must accelerate in order to leave Earth. Therefore, if they do not accelerate, they do not adopt a different frames of reference and they cannot be in uniform motion relative to one another they're all still just stuck on Earth.

If A and B's clocks agree, then either B accelerated from zero to light speed instantaneously, or A and B are not brothers, for B's point of departure is not Earth. If B's point of departure was not earth, then there is no basis for regarding A as occupying a privileged frame of reference or therefore as being "at rest."

Finally, in order for B to be moving away from Earth and C to be moving toward Earth at E2, C must have left Earth sooner or must have accelerated more; and in either of those cases, B and C's clocks would also not agree.

Also, in the Twin Paradox, I understand that while the traveling twin is decelerating for his turn and re-accelerating to return to Earth, the Earth-bound twin's time is dilated; as the traveler is making his turn, a long period of the Earth-bound twin's future life suddenly becomes part of the traveler's past without ever being present, which is what causes him to age less: he actually passes through less time than his brother. But how does one twin's aging compare with the other's during the traveler's initial acceleration away from Earth, or during his deceleration at the end of his return trip? Is it not the case that it is only as the traveler performs his u-turn that he ages more slowly? As the traveler initially accelerates during his departure from Earth, doesn't he age faster, and doesn't he also age faster as he decelerates for the last time? His frames of reference are changing at those times too. It should not go unnoticed that the combined amounts of acceleration away from the earth at both ends of the journey exactly equal the amount of acceleration toward the earth in the middle of the journey.

Dr. Craig, I had been looking for a biblically orthodox theologian who takes modern physics seriously for a long time when I found your commentary in the Moreland-Nielsen debate, "Does God Exist?" I suspect that cutting-edge science may someday clarify many biblical doctrines which seem otherwise problematic and intractable. Thank you for all you are doing.


Thanks for your kind remarks, Tom! I realize that Special Relativity may not be the most pressing concern on the minds of many of our readers, but in the interest of variety I picked your question to address this week.

The fundamental point of the so-called Twin Paradox is that Special Relativity, as Einstein formulated it, predicts that absolute effects will arise from merely relative motion. That is to say, these are not merely effects relative to a certain reference frame, as, for example, when A is shorter than B relative to frame R and B is shorter than A relative to frame . Rather here the effects are independent of one's reference frame; no matter which reference frame one picks, the traveling twin will be younger than his brother upon their reunion.

Now this is extremely bizarre when you think about it: how could merely relative motion, that is to say, motion which is such that either observer could be taken to be at rest and the other one in motion, result in absolute changes in only one of them? Philosophers and scientists who are not content simply to solve the equations and get on with the results but who reflect on the metaphysical implications of such a scenario are often deeply troubled by the fact that Einstein's version of the Special Theory predicts such absolute effects without any sort of dynamical causes. The results follow from the equations, but there is no physical cause of such absolute effects.

Now the Story of the Three Brothers is an attempt to rebut the objection that in the Twin Paradox the two brothers are not in merely relative motion: one of them is absolutely distinguished as the moving twin due to his acceleration and deceleration. Therefore, it should not be surprising if absolute effects arise as a result of his absolute motion.

The Story of the Three Brothers short-circuits this escape by eliminating the periods of acceleration and deceleration in the story. The three brothers simply pass each other at uniform speeds with no turnaround involved. The story that shows absolute, differential aging will arise as a result of their purely relative motion.

Now what you attempt to do, Tom, is expand the story in non-germane ways to reintroduce the acceleration and deceleration. But as a thought experiment, the story can be configured as we please, so long as it accords with the laws of physics. Thus, it just doesn't matter how the brothers got into the positions where they are; indeed, they needn't even be brothers--just three people of exactly the same age and appearance at the time of their initial respective rendezvous. All that matters is that their clocks, whether mechanical or biological, agree upon their initial coincidence. What the new story shows is that the aging of the traveling twin is not due to his absolute motion, much less to his acceleration or deceleration at various points during his journey, for these are entirely eliminated in the story involving three persons who merely pass one another in space.

Most theorists resolve the "paradox" by adopting a four-dimensional view of reality, such as was proposed by Herrmann Minkowski, which does away with reference frames and enduring three-dimensional objects in favor of shifting perspectives on four-dimensional objects in spacetime. But such a view, if taken metaphysically seriously, entails a tenseless theory of time which comes with a very high and, I think, unacceptable, price philosophically and theologically. Therefore, I cast my lot with H. A. Lorentz, who maintained that absolute time, absolute space, and absolute simultaneity do exist after all and the relativistic effects described in the Twin Paradox are due to absolute motion with respect to the privileged frame.