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John, in his spaceship traveling at a stable relativistic speed, is crossing the Milky Way in 500 years of his own time.
How many supernovae explosions would he experience?

From my reference frame on earth, his trip would take 100,000 years, and so, he should experience thousands of supernovae explosions, shouldn’t he?

But from his frame of reference, he’s seeing the Milky Way moving at relativistic speed around him and he does not feel that he is moving, which means that from his point of view, while himself aging of 500 years, he would see the Milky Way aging much slower (probably only a few years) and thus he would experience extremely few or no supernovae explosions at all, wouldn’t he?

What is the solution to the apparent contradiction in this thought experiment?

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    $\begingroup$ Does this answer your question? How can time dilation be symmetric? The answer, as pointed out there, is that the two observers are effectively measuring different starting and ending times for their trip. In your example, the number of supernovas are simply analogous to the ticks of a clock moving with the Milky Way. $\endgroup$ Commented Jun 18 at 16:49
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    $\begingroup$ Also see my answer here: Twin paradox without acceleration $\endgroup$ Commented Jun 18 at 16:52
  • $\begingroup$ @MichaelSeifert If the rate of supernovas are the tick of the clock, he should see less ticks and thus less supernovas during a mere 500 years, even less than someone see during 500 years at rest relative to the milky way as they explode at a slower rate to him. $\endgroup$ Commented Jun 18 at 21:15
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    $\begingroup$ It's curious that nobody mentions Doppler effect in these answers. What a moving observer SEES is NOT the Lorentz transformation. It's the Doppler effect. So whatever is in front of us speeds up enormously, whatever is behind us slows down. $\endgroup$ Commented Jun 18 at 23:00
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    $\begingroup$ It's supernova. Supernovae would be the plural, which is not correct here, see "car headlights". $\endgroup$
    – g.kertesz
    Commented Jun 19 at 7:45

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Let's draw space-time diagrams. First from your perspective on Earth:

enter image description here

This is hopefully straightforward to read. At time $t=0$ (lower green line) John (purple line) leaves Earth and heads for the delta quadrant, at a speed slightly slower than light (black dashed line), arriving at time $t=100,000$ (upper green line). Meanwhile, supernovae (orange circles) occur at various other stars in the galaxy (grey lines).

Now let's draw the same diagram, Lorentz transformed into John's reference frame. (This is guesstimated but should be qualitatively correct.)

enter image description here

This probably looks a bit weirder at first. But we can work out what's happening. First, John's path is now vertical, since he is not moving from his perspective. All the stars are now moving past him, at close to the speed of light, so they have become diagonal lines close to 45 degrees. The light path (black dotted line) is still at 45 degrees, because the speed of light is constant in all reference frames.

What makes it look weird is that the green lines have also tilted. In relativity, simultaneity is observer-dependent: things that happen at the same time in one reference frame can happen at different times in another reference frame, and if events are space-like separated they do not even have to happen in the same order in both frames.

We can add in additional lines for $t=0$ and $t=500$ in John's reference frame:

enter image description here

Now note that in the first diagram, there are 8 supernovae that occur between $t=0$ when John leaves Earth and $t=100,000$ when he arrives at his destination, while in the second there are only two supernovae that occur between $\tau=0$ when he leaves and $\tau=500$ when he arrives. So indeed, in John's reference frame there are fewer supernovae that occur during his journey than in yours.

But of course all 8 supernovae still occur in both diagrams. It's just that from John's perspective not all of them occur during his journey. Some of them occur before he leaves, and others occur after he arrives. This is the resolution to the paradox.

There are some subtleties about the above argument that can be appreciated with a bit of thought. The first is that, as several people have commented, John will not actually observe the supernovae until their light reaches him. To work out which supernovae he can observe, you can draw their future light cones and see if they intersect his world line. I leave this as an exercise to the reader. (If you do it on both diagrams you will find that my diagrams are not completely accurate. If they were, you would get the same answer for both diagrams.)

The second subtlety is that John's perspective will change when he accelerates and decelerates at the beginning and end of his journey. Although I didn't draw any, it is possible to have a supernova that occurs inside the yellow lines but outside the green ones. This means you can have a supernova that happens during John's journey in his reference frame, but when he comes to a stop, it will still be in the future. So from his perspective it happens twice! But he doesn't observe it twice, because its light will only reach him once, after he has arrived at the destination.

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    $\begingroup$ I don't get it because if he's leaving from Earth, then he have a quick acceleration to get to his 'almost speed of light' and then when he starts he's on Earth and those nova explosions that you mention 'in the past' , actually didn't yet exploded... Where are they gone? ... Would he experience suddenly a whole bunch of supernova explosion while he's accelerating for the very small acceleration phase of his trip? $\endgroup$ Commented Jun 19 at 17:14
  • $\begingroup$ @AlexandreBart I discuss that in the last few paragraphs. It is true that because of John's change of reference frame those events happen twice from his perspective. But the light from them only ever reaches him once, so he doesn't "experience" them twice. The meanings of "before", "after" and "at the same time" are a bit weird in special relativity: they're something you calculate after the fact, not something you directly observe. $\endgroup$
    – N. Virgo
    Commented Jun 20 at 0:46
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    $\begingroup$ In re "experiencing supernovae twice" aspect, this was discussed previously here $\endgroup$
    – Kyle Kanos
    Commented Jun 20 at 17:15
  • $\begingroup$ I am not sure this explanation makes sense. Modify the problem slightly so that the supernovas are in linear order and they explode locally as john passes them. How would your diagram look like in such a case? $\endgroup$ Commented Jun 21 at 0:15
  • $\begingroup$ In such a case he will see all the supernovas that explode in 100000 years of earth's time, but only 500 years of his own time. $\endgroup$ Commented Jun 21 at 0:22
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The problem is that you are ignoring the lack of synchrony between clocks, which compensates for time dilation. As one of the comments stated, if you look at multiple supernovas located at a given position in the galaxy, the rate of explosions the moving observer will perceive is very low, in fact slower than for an observer at rest relative to the galaxy.

But the supernovas are spread out across the galaxy. To help the visualization, let us imagine that the supernovas are arranged in order so they explode in sequence from the starting end of the trip to the end of the trip, so the traveler encounters the supernovas as they explode. The reason he will see many more supernova explosions than he would expect if only time dilation is considered, is that his clocks start out of synch with the clocks at rest with the galaxy. Using the lorentz transformations and assuming t=0 for the starting clocks as seen at rest with the galaxy, we see that the clocks for the moving observer are in synch with those at the beginning end of the galaxy, but out of synch by $\Delta t’=-\gamma*v/c*x/c$, with $v/c$ close to 1, and $x/c$ close to 100,000 years. So to him, the supernovas at the other end of the galaxy are supernovas from stars that are older by an amount of time of $100,000 \gamma$ years.

If all the stars were of teh same age for the observer at rest with the galaxy, to the moving observer instead, the farther stars will be way older than the close ones, which is the reason he sees more explosions than he should in 500 years. He sees old and young stars all exploding more or less at the same time, as to him the galaxy is composed of stars of different ages.

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – ACuriousMind
    Commented Jun 22 at 10:00
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The galaxy is about $100000$ light years across. If he crosses it so fast that time dilation makes only $500$ years pass on his clock, he is traveling at highly relativistic speeds. How fast is an easy calculation, but you don't need to do it. He is traveling at almost the speed of light.

This means that on our clocks, his journey will take about $100000$ years. Since a typical rate of supernova explosions in a galaxy is $1$ per century, about $1000$ of them will occur during his journey.

This is not the same as the number he will see. In our frame, light from supernovae behind him is not traveling much faster than he is. It will take a long time to catch him. His journey might well be over before it does. If he keeps on going (or stops), the light will catch him. So he will eventually see all of them.

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    $\begingroup$ This doesn't answer the question. $\endgroup$ Commented Jun 18 at 20:53
  • $\begingroup$ @VincentThacker The answer was updated... $\endgroup$
    – hft
    Commented Jun 18 at 21:50
  • $\begingroup$ @VincentThacker is right. This does not address the whole question. It addresses "experience thousands of supernovae". Since it improves the question rather than answer it, perhaps it should have been a comment. $\endgroup$
    – mmesser314
    Commented Jun 18 at 22:30
  • $\begingroup$ "He is traveling at almost the speed of light." If he's crossing 100000 light years in only 500 years, I'm pretty sure he's traveling faster than the speed of light. $\endgroup$
    – ouflak
    Commented Jun 19 at 11:06
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    $\begingroup$ @ouflak - In our frame of reference, which moves with the galaxy, we see him traveling at near the speed of light. Moving clocks run slowly. We see his clock measure $500$ years during the voyage. Our clocks measure $100000$ years. In his frame of reference, the galaxy moves past him near the speed of light. He sees his clock run normally, but our moving clocks running very slowly. The length of moving object are shortened. He sees the galaxy as $500$ light years across. His clock measures $500$ years during the voyage. $\endgroup$
    – mmesser314
    Commented Jun 19 at 14:00
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What does it mean to "experience" a supernova? I'm presuming this means to intersect the light-shell boundary that expands around a supernova.

From here on earth, we simply sit and wait for this light shell to reach us. If we assume on average one reaches us each century, then there are many of them already on the way (in our frame), but which we cannot yet detect.

The traveler will intersect many of these shells during the journey, with the same opportunity to gather information about the moment of the explosion as we have.

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Normally we would expect the supernova events to be randomly distributed in the galaxy, but that makes the analysis complicated, because what john actually sees depends on whether he is travelling towards or away from the events. Initially, let's just suppose the supernovae occur at the same location at the edge of the Galaxy at a frequency of once every 10,000 years in the earth reference frame. John is sent to investigate this strange region. The spacetime diagram for his journey is shown below::

enter image description here

The horizontal axis id distance and the vertical axis is time and both axes should be be multiplied by 10,000. From the diagram it can be seen that John sees 20 supernova events (the purple events) on his outward leg (the green worldline with a velocity close to c). Whenever he sees a supernova, he reports this back to Earth (the orange signal vectors). On arrival at Z, 500 years has passed according to John's clock and he sees the supernovae occurring with a frequency of once every 25 years. Earth calculates that John should arrive at Z after about 100,000 years earth time and in that 100,000 years Earth has only observed 10 supernovae. John has seen twice as many, essentially due to the Doppler effect due to moving towards where the signals are coming from. On arrival at Z, John sends a signal back to Earth stating he has seen 20 supernovae (the red signal vector), but Earth does not receive this information until 200,000 years have passed, by which time Earth will also have seen the same 20 supernovae events. If John turns around immediately at event z and returns to Earth at near the speed of light, he will not see any more supernovae from region Z, because he is outrunning the light from them and on his return he will have seen just as many supernova events as Earth did.

Now lets analyse what John sees if all the supernova events happened close to Earth (ignoring the damage to Earth).

enter image description here

In this new spacetime diagram John does no see any Supernova events on his outward journey (except $S0$ which occurred exactly as he was about to leave), because he is outrunning the light from those events. On arrival at Z he only seen one event and Earth has seen 10 supernova events at the time Earth calculates John has arrived at Z. After John turns around he sees all 20 events on his return trip and now he is seeing the supernova events occurring with a frequency of once every 12.5 years by his clock. (this is counterintuitive, if you expect John to see the evolution of the Galaxy as time dilated). John arrives back at earth shortly after the signal he sent from Z informing earth that he had not seen any events on his outward journey.

In both scenarios, John sees the same number of supernovae as Earth sees after completing a round trip and returning to Earth. It can reasonably be concluded that John will see exactly the same amount of supernova events as Earth sees, no matter where the supernova events are located in the galaxy.

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  • $\begingroup$ Thanks for this. I was tempted to make an attempt, but you have saved me that effort. With regard to "this is counterintuitive, if you expect John to see the evolution of the Galaxy as time dilated", I think that is the crux of the confusion in the question. When reviewing the twin paradox on wikipedia, the part that I think helps to understand is, on the return visit, the twins see events from each other passing more quickly, but if they account for the decreasing time delays, they calculate that time is passing more slowly for their twin. $\endgroup$
    – JimmyJames
    Commented Jun 24 at 14:41
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Impossible to define a solution because there are too many unknown variables and factors, ie. undefined, due to the ambiguity of possibility and observer's acuity and cognition.

But, the short answer is... an infinite amount or 0.

-_('.')_/-

Heh.

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Einstein came up with Special Relativity a few years before General Relativity. Later Einstein himself recognized that you can not look at SR alone. You must also take GR into account in these Twin Paradox cases. In particular you must consider the gravitational field that permeates the universe. He wrote here in 1918 "But all the stars in the universe can be thought of as taking part in the generation of the gravitational field... While in the special theory of relativity a portion of space without matter and devoid of an electromagnetic field is truly empty, it is quite different in general relativity. There, empty space has physical qualities mathematically characterized by the components of gravitation that determine the metrical behavior of that portion of space as well as its gravitational field."

He went on: "This situation can very well be interpreted by speaking of an ether whose state varies from point to point. However one has to be careful not to attribute to this ether any matterlike properties."

So this means that relative motion within the theory of SR, by itself, just doesn't work in the real universe. You also have to take the gravitational field into account.

So in answer to your basic question, John would see the same thousands of supernovae as you would. But he would see them pass very quickly in his time.

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    $\begingroup$ I totally agree with you that, we sometimes do have to keep in mind that SR is only a limit of GR, and if the GR corrections are important, we cannot just assume that SR alone is sufficient. However, this question is asked purely for constant relative velocity, and so there must be a valid answer purely in terms of SR that GR will also agree upon. That is why you are getting my downvote. $\endgroup$ Commented Jun 18 at 17:58
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    $\begingroup$ The paper you are referencing is so early on that Einstein did not yet know that his GR is actually in direct contradiction, i.e. disagreement, with Mach's principle, that he was trying to invoke. You are just wrong, albeit misled by Einstein. In any case, we have experimental proof that, taking into account GR and all, the experimental twin paradox agrees with predictions. You could have just learnt about them. $\endgroup$ Commented Jun 18 at 18:19
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    $\begingroup$ The only thing that separates GR from SR is the existence of gravity and spacetime curvature. SR itself is entirely capable of handling accelrations (although if you artificially and unneccessarily limit yourself to inertial frames of reference and Lorenz transformations, this isn't as clear). The standard twin paradox setup is completely absent any gravitionally relevant bodies, thus it is an SR phenomenon, and can be completely explained using only SR. $\endgroup$
    – Arthur
    Commented Jun 19 at 19:32
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    $\begingroup$ @foolishmuse What gravitational field? Two twins, out in open space, one drives away in their rocket, then turns around and comes back, while the other stays inertial. There is no need whatsoever for there to be an Earth where the trip starts. $\endgroup$
    – Arthur
    Commented Jun 19 at 19:52
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    $\begingroup$ Let's just take a moment to digest what @foolishmuse is arguing: The gravitational field is everywhere, and therefore you can't explain the twin paradox without it. Just as, for example, there are zoos in every major city, so it would be impossible to explain urban crime without mentioning zoos. $\endgroup$
    – WillO
    Commented Jun 20 at 1:40

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