The Twin Paradox
As indicated in the opening section, there are really 2 aspects to the twin paradox. The first relates to resolving whether the frames of reference in question are subject to asymmetric or symmetric time dilation. The second relates to how and where the time dilation in a given frame takes place. In principle, resolving which frame does not remain inertial, i.e. subject to no acceleration, is not really problematic, although the issue of an ‘absolute’ frame of reference will require some further discussion. As such, the actual paradox of which twin should be younger is not really a paradox, as the weight of authority is pretty well unanimous that it should be the space-faring twin who ends up being younger. However, the second aspect of explaining how and where time dilation takes place on-route is not as easily explained given that there are several approaches to consider:
While the list above is not exhaustive, it generalises a broad range of approaches that have been used to help explain the relative perception of time within each frame of reference under discussion. However, there is one omission that may be surprising, as it is the one adopted by Einstein in his 1918 paper based on his idea of General Relativity, which is the subject of another section in this website. However, the reason for not elaborating Einstein’s approach, at this point, is because adopting the arguments of general relativity seems to imply that special relativity cannot resolve the twin paradox. While the analysis of the twin paradox in terms of the equivalence principle in which general relativity links acceleration and gravitation is possible, the issue of relative time in each frame still appears to remain a matter of debate. However, there will be some further footnotes on the equivalence principle in the following discussion of ‘simultaneity analysis’.
Spacetime Diagram Analysis
So we will start by considering the implications that can be drawn from analysing 2 potential spacetime diagrams, as shown below. While these diagrams might appear to reflect a mirror-image under the rules of symmetric time dilation, only one of them is valid in the case of the twin paradox.
The spacetime diagram on the left represents the case where the Earth is considered to be the inertial frame from which the space-faring twin accelerates away in order to achieve a constant velocity of 60% light speed, i.e. v=0.6c. In fact, we might realise that there must be 4 periods of acceleration: the initial period of acceleration away from the Earth, the period of deceleration and acceleration at the mid-point [B] in order to turn round, plus the final period of deceleration on returning to Earth. However, it is worth noting that while special relativity puts an upper limit of velocity, i.e. [c], there is no upper limit on acceleration. As such, we might initially assume that the required acceleration for this journey takes place in almost infinitely small periods of time. While there are implications within this assumption, we will return to the details later in the discussion, but for now, we might simply assume that the presence of a G-force [F=ma] allows the rocket to be ‘unambiguously’ identified as the moving frame of reference.
On the assumption that the inertial and non-inertial frames have been identified, what can we say about time dilation?
Well, based on the spacetime diagram on the left above, we know that the Earth-bound twin should experience 10 years of elapsed time and, as a consequence of time dilation, the moving frame should only experience 8 years of elapsed time. For the moment, we shall simply accept that the reality of this difference in elapsed time can be verified at the end of the journey, based on the weight of evidence currently supporting special relativity. However, the identification of the Earth as the inertial frame immediately throws doubt on the second spacetime diagram, on the right above, because it is based on the assumption that both frames can be treated as symmetrical, i.e. the space-faring twin perspective can equally claim to be an inertial frame. However, given our assumption that the journey time for the space-twin can be verified to have taken 8 years, then the elapsed time on the Earth, moving with a relative velocity [v=0.6c], would have to be time dilated with respect to 8 years, i.e. only 6.4 years. Of course, this figure appears to conflict with the expectation that the earthbound twin should experience 10 years.
So can we draw any other spacetime diagram with respect to the moving frame?
We can, but we have to recognise that the moving frame represented by the rocket can never be shown within one single stationary frame. The easiest way to describe the spacetime diagram above is to start with the frame of the Earth twin, which now has to move away from the space twin now considered to be at rest with a velocity of [v=0.6c]. This initial phase is consistent with symmetrical time dilation for the outward phase of the journey to point-B, which the space twin would also have to view as moving toward him with a velocity of [v=0.6c]. However, at this point, the reality of asymmetric time dilation has to be resolved, because the Earth frame must continue to move with constant velocity [v=0.6c]. As such, the space twin has to ‘catch-up’ with the moving Earth by acceleration to a relative velocity of [0.88c]. So, in effect, we have created a third frame that is subject to further time dilation due to the higher relative velocity. However, the net effect is that the space and Earth twin are reunited after 10 years in the Earth frame, while only 8 years have elapsed for the space twin.
So what is the inference of the simultaneity gap on the diagrams?
The simultaneity gap is referring to the grey-shaded zones on the 2 previous diagrams, where the upper and lower dashed lines are known as ‘lines of simultaneity’ . While we have previously discussed some of the implications of these lines, they will require some specific analysis later in this discussion. However, if we look at the first set spacetime diagram, on the left, it can be seen that when the space twin reaches point-B, his perception of elapsed time corresponds to 4 years. However, the lines of simultaneity drawn from this point coincide with 2 different times in the Earth frame, which claim to be simultaneous in this frame, i.e. 3.2 and 6.8 years?
How can 1 point in time be simultaneous to 2 points in time in another frame?
Actually, the diagrams are misleading because they reflect the change in velocity from [v=+0.6c] to [v=-0.6c] to have taken place instantaneously. One potential implications of the simultaneity gap that has been suggested is that it corresponds to the actual time of deceleration and acceleration at [B], i.e. [6.8-3.2=3.6] years elapsed. Whether this statement can be supported will be taken up in the ‘simultaneity analysis’ discussion below.
Signalling Analysis
So, on the basis of the previous spacetime diagram analysis, we may feel there is a strong case for declaring that the Earth to be the inertial frame from which the space twin is seen to move with velocity [v=0.6c]. However, while this may then lead to the conclusion that this is an asymmetric time dilation system in which the space twin ages less than this Earth bound twin, the issue of relative time on-route remains.
How might we try to gain some insight of relative time on-route?
Well, the ability to send time-encoded light signals between the Earth-bound and space twin might prove useful. However, before we get into some of the details, let us consider the general situation, as shown in the diagram right, which may help to highlight the differences in the time signals caused by time dilation plus the propagation delay of the signal itself, due to the finite speed of light. In essence, this is the same spacetime diagram that we established in the previous discussion, which shows the inertial Earth frame and the frame of the space twin moving with a velocity [v=0.6c]. As such, our preferred reference will be the Earth frame and, as such; the line of simultaneity becomes a horizontal line in this frame. We can establish a known reference point at [B] by virtue of the Minkowski metric that might be considered as shorthand for the Lorentz time transforms in this case:
[1]
In order for a signal to arrive at [B], synchronised to this point in spacetime, it will require 3 years of elapsed time, as measured by the Earth frame, to propagate the 3 light-years to [B]. We might assume that this signal has its departure time encoded onto the light signal such that it can be decoded by the space twin, when he arrives at [B] after 4 years elapsed time in the moving frame. On receipt of this signal, the space twin immediately transmits his local elapsed time, i.e. 4 years, back towards Earth, which will take another 3 years according to Earth time, i.e. a total of 8 years. Now there are 2 ways in which to interpret the values in the signals:
- If the space twin was able to look back at a clock on Earth
through powerful telescope, when arriving at [B], he might think
that Earth time was running slower than his local space time, i.e.
2 years versus 4 years.
- Likewise, if the Earth twin was able to see a clock on the rocket, when arriving at [B], he might think the opposite in that Earth time would appear to be running faster than space time, i.e. 8 years versus 4 years.
However, there is not really any ambiguity in this situation as both the Earth twin and his space twin are both well aware of i) the effects of time dilation, ii) can both agree which is the inertial frame and iii) know the propagation delay due to the speed of light. It might also be worth pointing out that the relative speed [v=0.6c] is invariant in both frames of reference, which the space twin can calculate via the duration of the G-force measured onboard the rocket. All in all, both twins are capable of drawing and interpreting the previous spacetime diagram in terms of propagation delay and time dilation. As such, we can extend the previous diagram to show a series of light-based signals being sent in both directions, which would be subject to Doppler shifting to the red or blue end of the spectrum depending on the relative directions of the 2 frames, as shown below:
While the arrival rates of these signals may initially appear somewhat asynchronous in the receiving frame, they can be shown to reflect the consistent ticking of the clocks in both the Earth or space twin frames. Again, there are 2 factors at work, i.e. time dilation and propagation delay due to the finite speed of light [c]. The Lorentz transforms allow the time and distance travelled in both frames of references to be calculated and compared, and because the Earth frame is the inertial frame, the diagrams above will be resolved in terms of the distance measured in the Earth frame. The diagram on the left shows signals being sent from Earth to the space twin, while the diagram on the right shows signals being sent from the moving rocket back to Earth. It can be seen that some of the signals have been labelled, for illustration purposes, in the form X(Y), where [X] identifies the time in the source frame and [Y] the distance with respect to the Earth frame. As such, we can calculate the required time, with respect to the Earth frame, for the light signal [c] to propagate this distance, i.e. [t=d/c]. If we then remove this delay, caused by the propagation of the signal, the only discrepancy between the timing of the signals in either frame appears to be entirely due to time dilation. Therefore, again, there does not appear to be any paradox provided we anchor our interpretation to the inertial frame and consistently apply the Lorentz transforms
Footnote: To some extent, the reference to Doppler red and blue shifting was an unnecessary detail within this particular discussion, although it provides some useful background to the extension of the twin paradox that follows. However, it might be useful to note that ‘redshifting’, i.e. where the frequency or wavelength of an electromagnetic wave shifts up or down the EM spectrum range, can be caused by more than one reason. Where time dilation is involved, a relativistic Doppler shift has to account for all physical processes linked to the source, such as the atomic emission of photons, which are also slowed in time, which in-turn shifts the emitted light towards the red-end of the spectrum. In addition, there is gravitational and cosmological redshifting, where the former is also linked to time dilation effects due to present of a large mass-density and the latter linked to the expansions of the universe. |
Simultaneity Analysis
The first part of this description outlines some potential simultaneity effects within the twin paradox that do not really seem to make sense, but the idea is presented because it seems to illustrate some of the convolutions special relativity can cause . In the basic spacetime diagram analysis for our twin paradox example, the issue of the simultaneity gap was raised, but not really explained. However, it was suggested that the gap might reflect the rate of deceleration and acceleration at the turning point [B]. It was also highlighted that while, in practice, any period of acceleration or deceleration would take a finite time, in principle, these periods could be reduced to an infinitely short duration. In order to discuss some of the potential implications of this line of thought, the diagram on the right has superimposed a third frame of reference onto the original example. The purpose of the diagram is to provide two comparative frames in which the simultaneity gaps created by the lines of simultaneity are linked to turning points, e.g. [B] and [C], caused by the same change in relativistic velocity [v=±0.6c]. However, it might be noted that the relative span of each simultaneity gap appears to grow as a function of distance, which cannot be directly linked to the periods of acceleration and deceleration, if they are identical in both frames 2 and 3.
What inference might be drawn from the simultaneity gap?
Well, one suggestion forwarded is that the simultaneity gap spans a period of time in the inertial frame that is effectively compressed into the turn-round time in the moving frame. In the case of frame-2, this would imply that 3.6 years of time, i.e. 6.8-3.2 years, is compressed into the turn-round at year 4. While, in contrast, frame-3 would ‘see’ 7.2 years of time, i.e. 13.6-6.4 years, compressed into the turn-round at year 8.
How might this time effect be explained?
One initial line of thought might be that the periods of acceleration somehow align to the equivalence principle linking acceleration and gravity; as it is known that gravitation creates an additional time dilation effect. However, if this were the case, then the Lorentz transforms would not calculate the correct time dilation factor in isolation of general relativity and does not explain why identical periods of acceleration leads to different simultaneity gaps based on the distance travelled. Given these concerns, the following explanation of the lines of simultaneity is preferred because it seems to avoid the paradoxical perception of time implicit in the description above, which the previous signal analysis seemed to refute anyway. Let us step back and ask a question:
What is the scope of a line of simultaneity between 2 frames of reference?
In many respects, the line of simultaneity simply identifies 2 points in spacetime that are space-separated, i.e. they occur at the same time, in one given frame. In the spacetime diagram below, we are basically replicating the same twin paradox example used throughout this discussion. According to the inertial frame, the space twin travels at velocity [v=+0.6c] for 5 years, then ‘instantaneously’ turns around and travels back at velocity [v=-0.6c] for 5 years. At this turn round point, we see 2 lines of simultaneity linking frame-2 and frame-3 with frame-1. In some respect, the diagram above is replicating the previous discussion on ‘simultaneity’ by showing the axes of 3 frames of reference at [A]. The turn round point, at [B], is where the space twin switches from frame-2 to frame-3 and where we see the 2 lines of simultaneity linked to the 2 points on the time axis of frame-1, i.e. 3.2 and 6.8 years.
So, again, what inference might be put on these points?
Well, in many respects they are simply telling us that these points in the spacetime of frame-1 would be seen as simultaneous in time in frames 2 and 3 respectively. The first key point here is that the lines of simultaneity originate in two different frames of reference. So when the space twin first arrives at [B] in frame-2, the line of simultaneity is linked to frame-1 at 3.2 years; but when the space twin leaves [B] in frame-3, the line of simultaneity is linked to frame-1 at 6.8 years.
OK, but can we reconcile the meaning of simultaneity in these different frames?
Based on the current description, the 2 points, shown on the time
axis of frame-1, are 2 points separated by distance only in frames 2
and 3.However, when the space twin is in frames 2 or 3, he considers
himself stationary; therefore the arrival of a light signal from the
spacetime positions indicated by the lines of simultaneity should intercept the actual trajectory of the space twin moving with velocity [v=±0.6c]
at points that can be cross-checked with the idea of simultaneity. While
the actual calculation are not reproduced here, the shaded triangles
and the figures attached to the diagram above allude to the fact that
the simultaneity lines do correspond to a space separation, such that
we might again conclude that there is no paradox involved. However,
we shall defer drawing any further conclusions about the scope of the
twin paradox until after the next discussion, which extends this paradox
to encompass a similar journey undertaken by a set of triplets!