# Relativity of velocity entails equal velocity

1. Jul 18, 2014

### objecta99

its been years since I took a physics course so I might have what amounts to a really dumb question. However, I got into a group debate the other day where i was caught in the middle between person A who was claiming that the relativity of velocity did not entail the equal velocity of each frame, relative to each of only two frames and person B who claimed that it did. A spells this out thusly:

"... if the value of the velocity is a property of the relationship, then it follows that both have the same velocity (as the relationship is the same for both). If there are two objects in space, a rock, and a ship flying away from the rock (and no other reference point), if velocity is X for the ship, it's X for the rock per your view. You simply can't have it both ways. Either velocity is the relationship (rather than us needing the relationship to measure velocity), then it follow that whatever velocity the ship is moving at away from the rock, the rock is moving at the same velocity away from the ship (and hence time will slow down for the rock just as much as the ship). I'm saying you can't have it both ways."

I've been trying to find an explicit answer to this issue that doesn't just get too deep into the twin paradox. Essentially to rephrase, IF I were on a deserted planet in space and I launched into space with my ship, how do I know if I am moving away from the planet at some velocity or if the planet is moving away from me at the same velocity i'm traveling at. Person A was claiming that according to some "bad" interpretations of relativity theory that one would be forced to say that my velocity was the same as this planet's (even if we know that the planet obviously doesn't have the same velocity as the space ship at least per common sense).

I do not agree with this view, that is I disagree with person A's point of view on the matter but im not able to clearly say why without falling back on some uneducated vague ideas about the twin paradox. I am far, far from being proficient in physics. If anyone can help a bit here I would appreciate it a lot. I am way over my head on these matters especially as far as the math goes, so plz go easy on the math and any assumed understanding on my part:-) thanx.

Last edited: Jul 18, 2014
2. Jul 18, 2014

### Matterwave

Your description is a bit vague, but it sounds to me like person A is right. According to relativity, if the planet and spaceship are the only two reference frames, and are moving at a constant velocity, then it is equally valid to think of the spaceship as being stationary and the planet is moving away at speed v, as it is to think of the planet being stationary and the spaceship is moving away with speed v.

3. Jul 18, 2014

### objecta99

thank you, for your prompt reply. I apologize for any vagueness. To be clear (per my vagueness lol) Person A was arguing against the view that given only those reference frames then "the rock and the space ship" would have the same velocity relative to each other. Person B was holding that view explicitly and person A was claiming such view was incorrect. When I said I disagreed with said view (relative velocity entails equal velocity, relative to each of the two frames) I meant I disagreed with person A's criticisms of this interpretation. Though I betrayed myself a bit by qualifying my title with "This title is intentionally spurious as I do not endorse it". My first post and I have already come out the gate convoluted. I think I will edit it for the sake of clarity. That being said and to play A's devil advocate, how does one get around the intuitive common sense fact that when I leave Saturn in my space ship it seems hard to imagine that my velocity could be the same velocity of Saturn's JUST bc of velocity's relativity. I realize I'm begging a lot of SR gospel here perhaps, none of which im qualified to quibble with. Just trying for clarity on the matter. thanx again:-)

4. Jul 18, 2014

### Matterwave

In that case, person B is right.

There's just 2 objects moving relative (at constant velocity) to each other. Just 2 points of reference. There is no global, or universal, ether which can tell you which thing is moving and which is not. In object 1's frame of reference it (object 1) is still and object 2 is moving, in object 2's frame of reference it (object 2) is still and object 1 is moving. There's no judge that can tell you who is right and who is wrong.

Because we are used to having a stationary frame here on Earth (the Earth itself), we are not used to thinking of other things moving and us staying still. Specific effects on Earth, due to its gravity, and the presence of its atmosphere, tell us intuitively that WE are the ones moving because we can feel the wind against our cheeks, etc. But this intuition fails when it comes to relativity.

Think of it another way. What makes Saturn special? Why is not my ship just as special as Saturn. Which property of Saturn do you think tells you that really it's the SHIP that's moving and not Saturn? You will find that really, nothing makes Saturn special. The space ship is just as special. So there's no way to tell which one is REALLY moving.

5. Jul 18, 2014

### objecta99

Thank you very much:-)

6. Jul 19, 2014

### objecta99

I thought a bit about what I was trying to understand and i suppose it comes down to the issue of time dilation. I suppose my question then is, in what sense is time dilation relative if it is? Since we know that due to the frame change of space twin and the breaking of symmetry involved ultimately entails the fact that the space twin will age less *objectively or at least when back on earth how can time dilation be said to be relative? I suppose i am wondering if velocity is relative and if we cannot say which is moving away from what *objectivley, how do we say that time dilation is relative as well if we can tell who experienced the time dilation, as special relativity shows - and other experiments (muon concentrations etc). For example the jets clock is proven to run slower (and not the clock on the platform, in relation to the jet), gps atomic clocks are corrected for SR effects etc---these seem to me to argue that we do know which objects are undergoing time dilation not in a relative sense. Do experiments like the jet clocks or gps statellite clocks or muon concentration experiments get around the frame change symmetry breaking explination any better? Even if they do not it still seems that, though we can't judge which of two objects is undergoing time dilation at the time it's happening (as relative it appears both have "velocity" and that time slows for "both), we can make the temporal comparison after the fact (which we do) and understand that one had some velocity and the other did not (or had less velocity) by the virtue of only one haveing time dilation effects and the other not (younger twin). All this to say, how can we say time dilation is relative if we have these experiments that seem to allow us to tell which object is really undergoing time dilation effects?

to rephrase lol once again: Does the frame change account for the reason why time dilation is relative and not objective and does this frame change reason cover all other experiments (jet clocks, gps, muons) for why time dilation is NOT objective.

this might need to be rephrased, simplified and reposted as a new thread idk.

thanx again, im just trying best i can to understand this issue and any convolution is my own fault as i am not educated enough on the matter. thanx to you, any and all who can help.

Last edited: Jul 19, 2014
7. Jul 19, 2014

### ghwellsjr

Time Dilation is relative in exactly the same sense that velocity is relative because the formula for Time Dilation has velocity as its variable.

First off, you should discard the notions of "frame change" and "breaking of symmetry". They are completely unnecessary and will only create confusion.

Secondly, there are two separate issues going on here. The difference in aging between the earth twin and the space twin is not Time Dilation. We call the first issue Differential Aging and it is not relative. It can be calculated as the sum of all the Time Dilations for both twins over the entire scenario and will come out the same in all Inertial Reference Frames (IRFs) but different IRFs will involve different Time Dilations throughout the scenario. Time Dilation is an instantaneous effect while Differential Aging is an accumulation of all the Time Dilation effects during the entire scenario.

But once we select an IRF, we can objectively say who is moving and who is stationary.

If we pick the earth frame, we can objectively say that the earth twin remains stationary and the space twin is moving throughout the entire sceanario.

If we pick the IRF in which the space twin is stationary on the outbound half of his trip but then the earth will be moving during that time and when the space twin turns around, he will have to travel even faster in order to catch up with his bother.

As I just pointed out, by picking a different frame in which the earth twin is traveling, we can make both twins experience Time Dilation, it's just that the space twin ends up with more net Time Dilation added up during the entire trip than the earth twin.

I hope what I have already said clarifies these details. Let me go further with the twin scenario as seen by the two IRFs I mentioned earlier.

In the first IRF, the earth twin remains stationary while the space twin travels. Therefore, the space twin experience Time Dilation and ends up younger than the earth twin. Remember, the faster an object travels the greater the Time Dilation.

In the second IRF, the earth twin travels and experiences Time Dilation for the entire scenario while the space twin is stationary and experiences no Time Dilation for the first half of the scenario but then for the second half of the scenario, the space twin has to turn around and travel even faster than the earth twin in order to catch up to him. This higher speed results in more Time Dilation than the earth twin experiences so that when the re-unite, the accumulated effects of Time Dilation for both twins results in the space twin aging less than the earth twin by exactly the same amount as the first IRF calculated.

Remember, stick to one IRF for the entire scenario--no frame changing, and calculate the instantaneous Time Dilation based on the velocity in your chosen IRF and add up the accumulated times for all participants.

Does that help? Any more questions?

8. Jul 19, 2014

### Staff: Mentor

Don't confuse time dilation with the differential aging that you see in the twin paradox. They're related, but they're not the same thing.

If A and B are in motion relative to one another, then A's clock will run slow in a frame in which B is at rest and B's clock will run slow in a frame in which A is at rest. This situation is completely symmetrical, and there's no way in which we can say that one of them is "really" moving and the other is not. That's time dilation.

Time dilation is symmetrical even in the twin paradox - at every moment of both the outbound leg and the inbound leg of the space twin's journey, the earth clock is slow relative to the space twin's clock and vice versa. The space twin ends up less aged than the earth twin because he took a different path through spacetime than did the earth twin, the two paths have different lengths, and the path with the turnaround in it is objectively the one on which less time passes.

You might try thinking about the symmetrical variant of the twin paradox. Here the two twins are both in twin spaceships, they travel in opposite directions for a while, then turn around and travel back to their starting point. If the situation is exactly symmetrical (both traveled at the same speed, both travelled the same distance) they will both be the same age when they meet up again - yet there was time dilation between their clocks at every step of the way.

9. Jul 19, 2014

### objecta99

Thanx for taking the time to walk me through this. This was very helpful especially the difference between time dilation and aging differentials—something which I can only vaguely grasp at the moment. Am I understanding aright that what's crucial to the space twin aging less is his/her turning back to earth and thus having to out-speed the earth twin in order to catch up—thus adding up with less time. So its during the catch up relative to the earth-bound space twin that the extra velocity causes the space twins comparatively more 'youthful' arrival when we add up?

If that is right and if I haven’t done a hatchet job on your explanation then I have two follow up question--thingys:

1)Once the full trip has been made by the space twin and he/she rejoins their sibling and realizes, “hey you have aged more!” can we say that by that fact we know “objectively” that the space twin has traveled at some point at a greater velocity relative to their twin? Then does that imply that we can say that of the two ppl one has traveled faster (objectively) than the other? This would, I presume (and based on what Nugatory has said) be bc of the different paths in space/time taken and not bc the space twin experienced more time dilation? Am I parsing the dilation and accumulated aging distinction right sort of? Does the path difference involve curve vs straight line stuff due to velocity? IS it the angular difference between the two paths that is objectively different, is it space contraction--does this have any relationship to curved space/time (bone head question probably). sorry to go all over the place here but if its not time dilation that's the essential difference but the aging differential what property of those two paths accounts for that ageing differential beyond stating that their is a added up time differential or that in spacetime the straight-line path would always be the shorter one no matter how the bendy path was drawn in any coordinate system.

2) Is the evidence for time dilation as shown with SR effects on muon concentrations and the experiments with 'slow' clocks on jets, saying anything different about how we understand time dilation compared to the twin paradox? IOW do these experiments offer any support to the idea that time dilation is not relatively symmetrical as in the case of the twin paradox.

As a caveat, I know I probably sound a bit 'slow' on the up-take here but the context for my question is in regards to a group debate on ethical realism vs ethical relativism. So that's why I am trying to sort of play devil's advocate for “time dilation is not relative”, plus I couldn’t pass an intro to physics test if one was given to me right now. Of course in any philosophical debate analogies and references are made to SR and GR and so I'm trying to straddle the two different contexts. I personally accept that time dilation is relative (like velocity) but this other guy (who just published a book on free will) is claiming that time dilation is not relative due to the age differential in the twin paradox (which we can verify after the fact or after the trip) and due to the experiments on muons and jet carrying clocks. This guy wants to say that bc of the age differential we can say that time dilation is not AS relative as often assumed or iow but a bit different, by knowing the age differential effect we can objectively ascertain which twin has traveled faster. I've had to outsource to my scientific betters (betters would be an understatement) in order to better understand whats going on with the relativity of time dilation. Thank you for taking the time to clarify these matters for me I really, really appreciate it and take care-)

Last edited: Jul 19, 2014
10. Jul 19, 2014

### Staff: Mentor

This is certainly false reasoning. The fact that two entities have some relationship does not imply that the relationship is the same for both. I may have a relationship with someone, and my relationship with that person may be that I am his father, but if that is the case then he certainly is not my father.

11. Jul 19, 2014

### Staff: Mentor

I am not sure what "objectively" means in this context but what you can say is that in any case where two clocks begin together, separate, and reunite later and where the spacetime is flat (no gravity) then the one that aged less during the separation must have traveled faster at some point relative to any single inertial frame.

Regarding the property of the path, that would be the "length" of the spacetime path, known as the spacetime interval: $ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2$. The path with the greater timelike length is the one that has the greater aging. This statement holds even if gravity is significant or non-inertial frames are involved, and the particular value of the interval for each path is frame invariant.

All of that evidence is consistent with relativity.

12. Jul 19, 2014

### A.T.

Ethical relativism has nothing to do with the theory of relativity in physics. Just because something has the same or similar name, doesn't make it related. It's just a name that stuck, but might have just as well end up different:

http://en.wikipedia.org/wiki/Criticism_of_relativity_theory#Philosophical_criticism

13. Jul 19, 2014

### phinds

objecta99, one of the things that has not been explicitly discussed in this thread is space time paths and that's something you might find it interesting to look into. One way of looking at the Twin "paradox" is that although when they are first together, that can be thought of (a bit broadly) as an event in space-time and when they meet up again when the traveler gets back, that's another even in space-time. What happen in between is that they take different paths through space-time to get from one event to the other. One of those paths involves more distance and less time than the other. I found this point of view to be helpful when I first started reading about this stuff.

14. Jul 19, 2014

### objecta99

A.T.: As a matter of philosophical opinion you are entitled to it. As a matter of philosophical 'fact' you might be right and I don't necessarily disagree with you as I have personally argued against the analogy for precisely the reason you highlight with that wiki quote (Lorentz invariance and speed of light constant etc). As a matter of the philosophical literature there are more than several precedents where the analogy has been made by moral relativists re: Bernard Williams and Gilbert Harman.

15. Jul 19, 2014

### Staff: Mentor

No, that's not quite how it works. We have two paths through spacetime, and one of them is shorter than the other. It makes no sense to talk about where the extra time is added to the longer path; every step along a path contributes to its total length.

I also suggest that you not take on the twin paradox until you're very clear in your mind how ordinary symmetric constant-relative-speed time dilation is free of paradox. A and B are moving relative to one another, A says that B's clock is running slower than A's, B says that A's clock is running slower than B's, and they're both right. How can this be? How is it logically possible for both clocks to be slower than the other? You have to understand this case solidly before you can take on the more complicated twin paradox.

Last edited: Jul 19, 2014
16. Jul 19, 2014

### Staff: Mentor

Although you might luck into the occasional counterexample, you could reasonably assume that all such analogies and references are bogus unless they're made by someone who has a fairly solid physics background.

17. Jul 19, 2014

### objecta99

Thanx Nugatory, feel free not to respond if you find my level of understanding not satisfactory as I have repeatedly admitted. but all the same thanx for pointing out that showing where the extra time came from is not necessary and its all in the path length. have a nice day;-)

18. Jul 19, 2014

### Staff: Mentor

You're doing fine here... The expected outcome of this discussion is that you will become one the good counterexamples that I alluded to.

19. Jul 19, 2014

### ghwellsjr

Keep in mind that I said Time Dilation is relative just like velocity is relative to an Inertial Reference Frame. In the first IRF that I mentioned in post #7, the earth twin's rest frame, the space twin is not "catching up" to the earth twin at any point since the earth twin is at rest. The space twin is merely going back home. It's in the second IRF where the earth twin is the one that is traveling throughout the scenario that the space twin, after remaining at rest for some period of time has to travel faster than the earth twin to catch up.

Perhaps some spacetime diagrams for a concrete example will help to make it clear. I will use the example of the space twin traveling away from the earth twin at 60% of the speed of light for four of his years and then turning around and coming back to the earth twin at the same speed, as defined by the rest frame of the earth twin. The earth twin is depicted in blue and the space twin in red. The dots mark off one-year increments of time. If the dots correspond to the Coordinate Time, then there is no Time Dilation. If the dots are stretched out farther than the Coordinate Time, then there is Time Dilation. Here is the diagram for the earth's rest frame:

As you can see, the earth twin has aged 10 years and the space twin has aged 8 years (count the dots, starting from zero).

Now we can transform the coordinates of all the events (dots) in the above diagram to an IRF that is traveling at 60%c with respect to the first IRF. This will make the space twin stationary for the first half of his "trip" (which is not really a trip yet since he starts out stationary in this IRF):

This is the situation where the space twin has to travel at faster than 60%c to catch up to the earth twin.

Please understand that both of these diagrams depict the exact same scenario, we have just looked at it from two different IRF's which change the Time Dilations during the three straight line segments (one in blue, two in red) but the aging of each twin remains the same (10 for blue, 8 for red). Note that blue ages by the same amount during the entire scenario but red ages faster at the beginning while he is at rest and slower while he is traveling to catch up.

Now I'll show you the IRF in which the space twin is at rest during the last half of his "trip":

Now the space twin is not catching up, but rather slowing down to a stop in order to let the earth twin catch up to him.

Next, I want to show you another IRF, just to make sure you don't have the idea that the only valid IRF's are those in which an observer is at rest. In this IRF, we start with both twins traveling in opposite directions at the same speed and then the space twin turns around to catch up to the earth twin:

Note that both twins are Time Dilated by the same amount at the beginning of the scenario and then the space twin ends up Time Dilated even more so that their ages are the same as in the other IRF's.

And we can do another IRF in which the twins start out traveling away from each other with the space twin going faster but then when he turns around, they approach at the same speed:

Can you see how in each of these IRF's, the speeds of the two twins are different and so the Time Dilations are different but when you add up the amount of aging for each of them, they end up the same?

For the simple case where one twin remains inertial (doesn't accelerate) and the other twin does accelerate, then you can make statements like this but I don't think it is useful to think that way because there are other scenarios where both twins accelerate and for which the statement it is not true that the one that traveled at some point at a greater velocity is the one that ends up younger.

For example, here is a scenario where one twin starts off traveling in one direction at 80%c for a year and then returns at a slower speed while the other twin travels all the time at 60%c:

The first twin ages 9 years while the other one ages 8 years. So it is not true in general that you can just identify the younger twin simply by noting which twin traveled faster. And remember that speed is relative to the IRF so you can change which twin travels faster (in some scenarios) simply by transforming to a different IRF. Shortcuts and "rules of thumb" tend to be dangerous in SR. Better to just work out the details of any scenario (using any IRF you want) to see what happens.

I think I have given you enough for you to figure out the rest of your questions.

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20. Jul 19, 2014

### objecta99

this was very helpful, I got tripped up on the phrase "catch up" and now I see what you were talking about. I was assuming just the first diagram where one was at rest relative to the other and vice versa the whole trip. now I see what u were saying about the catch up thanx a million. I have a lot of homework to do.