Introduction
The kinematics of Special Relativity (SR) is predicated on the Lorentz transforms [Lorentz1952]. Although these equations are merely (rescaling) transforms that conserve the covariance of physical laws across relatively moving inertial frames, it is the general perception that the effect of time-dilation (derived from the Lorentz transforms) is a manifest physical effect [1]. As any student of SR knows, this interpretation leads to various paradoxes; notably, the Twin (or Clock) Paradox which is the focus of discussion here [2].
The Twin Paradox
Very simply, the Twin Paradox involves two identical twins, one of whom stays on Earth (frame O) while the other, an astronaut (frame O'), goes on a space-faring journey in a high-speed rocket. Upon her return, the astronaut twin ostensibly finds that she has aged less than her sister who stayed back on Earth - presumably due to effects of time-dilation. This is clearly a paradox because the astronaut twin could very well argue that it's her sister on Earth who should age less, due to the very same effects of time-dilation.
The paradox has been explained away in multiple ways, but mostly by invoking the fact that the situation is not completely symmetrical. For instance, only the astronaut twin actually experiences acceleration when she turns around. Alternatively, the astronaut twin uses two inertial frames - one in each direction of travel - and it's the switch between the frames that causes the asymmetry [Schutz1985].
I submit that all the explanations of the Twin Paradox are faulty; primarily because they seek to explain something that is not a paradox in the first place. And to demonstrate this, I have to re-frame the problem in a way that eliminates the underlying basis for asymmetry.
A Cylindrical Minkowskian Universe
Imagine a flat two-dimensional universe. By flat, I mean a universe with an intrinsic Minkowskian metric. While the intrinsic metric is Minkowskian, SR doesn't place (nor can it place?) a restriction on the extrinsic topology of a Minkowskian universe. As a result, it is perfectly legitimate to consider the case where a Minkowskian universe is extrinsically curved onto itself. In particular, assume that the x coordinate axis of O wraps around to meet itself. The topology of this universe then is that of the surface of a cylinder. The y coordinate axis lies along the length of the cylinder while the x coordinate axis encircles the girth of the cylinder.
Re-framing the Twin Paradox
Now, reconsider the Twin Paradox in a cylindrical Minkowskian universe: The astronaut twin, who sets off in the x direction of the twin on Earth, will eventually 'circumnavigate' the universe and rendezvous with her Earth-bound sister. Under these circumstances, there is no switch of inertial frames nor is there any acceleration. The situation is completely symmetrical, and the usual explanations of the Twin Paradox no longer apply. Thus, under the conventional interpretation of time-dilation as a physical effect, we are once again faced with the paradox of determining which twin has aged more than the other.
Reinterpreting time-dilation
The Lorentz transforms have been verified empirically over and over again. Provisionally, one can then assume that the transforms are correct. Since we nevertheless arrive at a paradox above, the only conclusion is that the physical interpretation of time-dilation in SR is to blame [3].
Since the situation is perfectly symmetrical, the one thing we can be certain about is that neither twin is older than the other. Alternatively put, they've both aged the same amount of time physically.
In other words, the effect of time-dilation derived from the (rescaling) equations of the Lorentz transforms does not actually imply slow down of physical time in a moving inertial frame. The Lorentz transforms are, simply put, rescaling equations that achieve covariance of physical laws across inertial frames. In effect, both O and O' physically age equally. However, if we wish to maintain the form of the laws of physics as we move from frame O to frame O', we need to employ scaling in the units used to express the laws. This scaling is a purely kinematical (and not physical) artefact and it is incorrect for us to infer any physical results from it.
Other Notes
Since my initial explorations on this topic I've come across multiple other expositions that argue that time-dilation is not a physical effect: for instance Kracklauer [Kracklauer2002], and most notably multiple references to Sachs [Sachs1971] (which I've not been able to obtain a personal copy of). However, to the best of my knowledge, no one has used the formulation of the Twin Paradox in a cylindrical Minkowskian universe as stated above, and I believe it is my unique contribution to this problem.
Footnotes
[1] That such a physical effect should arise simply from kinematical considerations is untenable. Unfortunately, even Einstein initially believed time-dilation to be physically true: “... it follows that the time marked by the clock (viewed in the stationary frame) is slow ...” [Lorentz1952].
[2] The Ehrenfest paradox similarly arises from interpreting length-contraction as a manifest physical effect.
[3] Alternatively, it is possible that SR prohibits extrinsically curved universes whose intrinsic metric nevertheless remains Minkowskian. However, there is nothing in SR that compels us to assume this at this time.
Update [20Sep2009]: I've emailed a précis of these deliberations to Dr. Mendel Sachs via his website.
The kinematics of Special Relativity (SR) is predicated on the Lorentz transforms [Lorentz1952]. Although these equations are merely (rescaling) transforms that conserve the covariance of physical laws across relatively moving inertial frames, it is the general perception that the effect of time-dilation (derived from the Lorentz transforms) is a manifest physical effect [1]. As any student of SR knows, this interpretation leads to various paradoxes; notably, the Twin (or Clock) Paradox which is the focus of discussion here [2].
The Twin Paradox
Very simply, the Twin Paradox involves two identical twins, one of whom stays on Earth (frame O) while the other, an astronaut (frame O'), goes on a space-faring journey in a high-speed rocket. Upon her return, the astronaut twin ostensibly finds that she has aged less than her sister who stayed back on Earth - presumably due to effects of time-dilation. This is clearly a paradox because the astronaut twin could very well argue that it's her sister on Earth who should age less, due to the very same effects of time-dilation.
The paradox has been explained away in multiple ways, but mostly by invoking the fact that the situation is not completely symmetrical. For instance, only the astronaut twin actually experiences acceleration when she turns around. Alternatively, the astronaut twin uses two inertial frames - one in each direction of travel - and it's the switch between the frames that causes the asymmetry [Schutz1985].
I submit that all the explanations of the Twin Paradox are faulty; primarily because they seek to explain something that is not a paradox in the first place. And to demonstrate this, I have to re-frame the problem in a way that eliminates the underlying basis for asymmetry.
A Cylindrical Minkowskian Universe
Imagine a flat two-dimensional universe. By flat, I mean a universe with an intrinsic Minkowskian metric. While the intrinsic metric is Minkowskian, SR doesn't place (nor can it place?) a restriction on the extrinsic topology of a Minkowskian universe. As a result, it is perfectly legitimate to consider the case where a Minkowskian universe is extrinsically curved onto itself. In particular, assume that the x coordinate axis of O wraps around to meet itself. The topology of this universe then is that of the surface of a cylinder. The y coordinate axis lies along the length of the cylinder while the x coordinate axis encircles the girth of the cylinder.
Re-framing the Twin Paradox
Now, reconsider the Twin Paradox in a cylindrical Minkowskian universe: The astronaut twin, who sets off in the x direction of the twin on Earth, will eventually 'circumnavigate' the universe and rendezvous with her Earth-bound sister. Under these circumstances, there is no switch of inertial frames nor is there any acceleration. The situation is completely symmetrical, and the usual explanations of the Twin Paradox no longer apply. Thus, under the conventional interpretation of time-dilation as a physical effect, we are once again faced with the paradox of determining which twin has aged more than the other.
Reinterpreting time-dilation
The Lorentz transforms have been verified empirically over and over again. Provisionally, one can then assume that the transforms are correct. Since we nevertheless arrive at a paradox above, the only conclusion is that the physical interpretation of time-dilation in SR is to blame [3].
Since the situation is perfectly symmetrical, the one thing we can be certain about is that neither twin is older than the other. Alternatively put, they've both aged the same amount of time physically.
In other words, the effect of time-dilation derived from the (rescaling) equations of the Lorentz transforms does not actually imply slow down of physical time in a moving inertial frame. The Lorentz transforms are, simply put, rescaling equations that achieve covariance of physical laws across inertial frames. In effect, both O and O' physically age equally. However, if we wish to maintain the form of the laws of physics as we move from frame O to frame O', we need to employ scaling in the units used to express the laws. This scaling is a purely kinematical (and not physical) artefact and it is incorrect for us to infer any physical results from it.
Other Notes
Since my initial explorations on this topic I've come across multiple other expositions that argue that time-dilation is not a physical effect: for instance Kracklauer [Kracklauer2002], and most notably multiple references to Sachs [Sachs1971] (which I've not been able to obtain a personal copy of). However, to the best of my knowledge, no one has used the formulation of the Twin Paradox in a cylindrical Minkowskian universe as stated above, and I believe it is my unique contribution to this problem.
Footnotes
[1] That such a physical effect should arise simply from kinematical considerations is untenable. Unfortunately, even Einstein initially believed time-dilation to be physically true: “... it follows that the time marked by the clock (viewed in the stationary frame) is slow ...” [Lorentz1952].
[2] The Ehrenfest paradox similarly arises from interpreting length-contraction as a manifest physical effect.
[3] Alternatively, it is possible that SR prohibits extrinsically curved universes whose intrinsic metric nevertheless remains Minkowskian. However, there is nothing in SR that compels us to assume this at this time.
Update [20Sep2009]: I've emailed a précis of these deliberations to Dr. Mendel Sachs via his website.
References
[Kracklauer2002] Kracklauer, Kracklauer, "Twins: never the twain shall part", Proceedings: Physical Interpretations of Relativity Theory VIII, 2002, pp. 248-255.
[Lorentz1952] Lorentz et al., "The Principle of Relativity", Dover Publications, 1952.
[Sachs1971] Sachs, "A resolution of the clock paradox", Physics Today, vol. 24(9), 1971, pp. 23-29. http://dx.doi.org/10.1063/1.3022927
[Schutz1985] Schutz, "A first course in general relativity", Cambridge University Press, 1985.
