This paradox is very common it seems, in which A sees B's clock running slow and B sees A's clock running slow. Here is the question a little more concretely.  Let's say B flies by A's spaceship. If B's clock advances $t$, then in that time A's clock advances $\gamma t$ through time dilation (This is easily shown if there is a light clock next to B). Now from A's perspective (assume there is a light clock next to A), during that same time interval, he sees B's clock run slower and thus this time interval must be $\gamma^2 t$. Equating the time intervals gives $t = \gamma^2 t$ and $\gamma = 1$, which is clearly wrong. How is this reconciled?
2335
I read a lot about the classical twin paradox recently. What confuses me is that some authors claim that it can be resolved within , others say that you need . Now, what is true (and why)?
49355
I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!
13670
While answering the question  it occurred to me that time would run more slowly at the equator than at the North Pole, because the surface of the Earth is moving at about 464 m/s compared to the North Pole. The difference should be given by:  $$ \frac{1}{\gamma} \approx 1 - \frac{1}{2}\frac{v^2}{c^2} $$  and at $v = 464\ \mathrm{m/s}$ we get:  $$ \frac{1}{\gamma} \approx 1 - 1.2 \times 10^{-12} $$  This is a tiny difference – about 4 days over the 13.7 billion year lifetime of the universe – but  the accuracy of current atomic clocks is about $1$ part in $10^{14}$, so the difference should be measurable. However, I have never heard of any measurements of the difference. Is there a flaw in my reasoning or have I simply not been reading the right journals?
113155
This is related to  and  The idea is to shine a beam of light from a clock towards a heavy mass, like a black hole, in some manner such that the curvature of spacetime around the mass is sufficient to &quot;bend&quot; the light directly back to the clock. That is, to send the light along a self-intersecting geodesic using a sufficiently curved spacetime. Note that in this case the source and the detector are the same object, so no clock synchronization is required. Hypothetically, this should measure the one-way speed of light. None of the given refutations from the similar posts on this site are convincing. Here are some of the main refutations, as I understand them, and my counter to them: (1) In being curved by spacetime, the light is actually changing direction, hence this is not the one-way speed. To my understanding of General Relativity, the light travels along a geodesic; that is, a curve such that, ignoring technicalities of Differential Geometry, its tangent direction is constant. There is no reflection or acceleration. So in this sense, how can the light with a constant tangent direction be said to change direction? One might say that the path deviates from a straight line in Euclidean space, but given that spacetime is not Euclidean, has no , and has no global notion of direction, this is the only definition of straight or &quot;one direction&quot; that makes sense. Thus, either the light does not change direction and is &quot;one-way,&quot; or we need to be clearer about what exactly we mean by &quot;one-way&quot; in the context of General Relativity. (2) Such an experiment would be impossible or impractical. I think some posters were saying this because the original questions implied that the clock would have to be at the event horizon of a black hole, which is of course impossible or impractical; however, I don't think such an extreme closeness to the black hole is required. Given that the  of a black hole, where light orbits in a circle, is outside of the event horizon, shining the light at some distance outside of the photon sphere should be sufficient to curve the light back to the clock, regardless of the distance of the clock from the black hole (see my rough illustration). At the extreme, we could place the clock directly on the photon sphere, which should be theoretically possible. Regardless, the question is not about practicality, but whether the one-way speed is theoretically measurable.  Update: Essentially what this argument boils down to, I think, is that the average speed of light along a single self-intersecting geodesic may differ from the average speed of light along a non-geodesic self-intersecting path, which is always c. Put that way, however, it does seem a little ridiculous. It is hard to see how the anisotropy of the speed could be defined in such a way as to be consistent with this pathological case. Presumably, we could approximate a single self-intersecting geodesic with a non-geodesic path arbitrarily closely, so, logically, the average speeds over each path should be approximately the same, i.e., c. Furthermore, how would the &quot;preferred direction&quot; of the speed of light behave around a black hole, for instance? Regardless, I still think this experiment does actually solve the problem for the given notion of &quot;direction&quot;. Though the problem is usually posed in the flat spacetime of Special Relativity, with a global notion of direction, so the apparent unanswerability of this question should be seen as a limitation of Special Relativity only, which is then solved by General Relativity.
24098
Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric.
79236
This is a quote from the book A first course in general relativity by Schutz:     What we shall now do is adopt a new unit for time, the meter. One meter of time is the time it takes light to travel one meter. The speed of light in these units is    $$\begin{align*}\end{align*}$$      $$\begin{align*}  c &amp;= \frac{ \text{distance light travels in any given time interval}}{\text{the given time interval}}\\ &amp;= \frac{ \text{1m}}{\text{the time it takes light to travel one meter}}\\ &amp;= \frac{1m}{1m} = 1\\ \end{align*}$$      So if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!   Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this:  The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$  Does Schutz correctly show how $c$ can be a dimensioness constant?
52998
The FRW metric is given by: $$ds^2=-dt^2+a^2(t)\ dr^2$$ where $ds$ is an interval of proper length, $dt$ is an interval of cosmic time, $dr$ is an interval of co-moving co-ordinate distance and $a(t)$ is the scale factor (also $c=1$).  If I take $dt=0$ then I find that an interval of proper distance $ds$ is given by: $$ds = a(t)\ dr$$  Thus the proper distance between two nearby co-moving points is proportional to the scale factor - space expands.  If I take $ds=0$ then I obtain the null geodesic equation describing the path of a light ray: $$dt = a(t)\ dr$$  Thus the light travel time between two nearby co-moving points is also proportional to the scale factor.  Does this imply that intervals of cosmic time expand along with intervals of space?  The natural clocks in co-moving co-ordinates are expanding light-clocks.   Maybe in order to derive the constant units of the "atomic" time that we experience, $d\tau$, from expanding cosmic time intervals $dt$, we need to use the conformal time equation: $$d\tau = dt/a(t)$$
23658
Is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then to B it APPEARS that A's clock moving slower, but A sees his own clock moving at normal speed.  Similarly, to A it APPEARS that B's clock is moving slower, but B sees his own clock moving at normal speed.  If the above is true, then both A seeing his own clock moving at normal speed and B seeing his own clock moving at normal speed means that in reality both clocks are moving at normal speed, and neither has slowed down, whereas the other person's clock APPEARING to move slowly is merely an illusion.   Now is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then A sees his own clock move slowly (compared to the speed of the clock when A was at rest with respect to B) and B sees his own clock move slowly, so that in reality both clocks are moving slowly, but they still remain synchronized (since both are slow by the same amount)  If the above statement is not true, then why do muons decay slowly when moving fast? [it could only be possible if the muon saw its own clock as moving slowly. If we saw the muon's clock moving slowly, but the muon saw its own clock moving at the normal rate, then the muon would decay at the normal rate, and not slowly]  Can anyone please explain where I went wrong?
58833
Can you resolve this paradox: Time and space are supposed to be treated symmetrically in SR, yet time dilates while length contracts. I've been working on this for weeks, and I can't resolve it.  The mirror experiment shows that time and space both expand for the moving observer, in contradiction to the results one derives from the LT, yet a well-known text, Eisberg's Fundamentals of Modern Physics (pp. 20-21) incorrectly concludes the usual asymmetrical results from that, by changing the definition of time dilation! He defines it as meaning that time measured in the moving frame is longer, when tau is actually well-known to be shorter. Time dilation refers to the mover's time UNITS LOOKING longer to the stationary observer because the total number of time units is FEWER, the opposite situation from the mirror experiment.
77492
I am trying to understand the Twins Paradox (relativity) and its implications.  Can we infer from the Twins Paradox that the ageing process (cell decay or other biological processes) depends on motion? When I move fast, will my rate of ageing slow down?  I know the theory says that its time which slows down but my understanding (it may be wrong) is that time is just a human perception, it isn't a real physical entity. If time isn't real, outside human perception, then it must be the biological process of ageing being impacted by motion, in the Twins Paradox experiment.  Please help me, what am I getting wrong?
91613
I'm not familiar with any complicated physics equation, however I do understand some basics. Suppose there is two objects, both of them are moving away from each other in a 3-dimensional space, which they both have the speed of half the speed of light ($c/2$). Relative to the reference frame of object A, object B would be moving away from itself equal to the speed of light.     A          B &lt;-----      -----&gt;   -c/2       c/2   to a 3rd observer    0          c    to A   -c          0    to B   This seems to break the law of "nothing can move faster than the speed of light". I believe that there is some other explanation for this case. I have done some research, and there are quite a few questions on this topic. Take  and  for example. However, they are about adding velocities together, which isn't quite the same as in the case I was describing.   is also similar to my case, however the two objects described are both moving toward the same direction, and I'm not shooting any beam or laser toward the other object.  Most probably in reality there are some extremely complex laws and equations which makes this question more complicated. However, the two objects are not interacting with each other and I'm not trying to "detect" the speed of the other object in the reference frame of A or B. Also none of the objects are moving at a significant fraction of the speed of light relative to the third observer, so special relativity doesn't apply(?) I just could not think of any reason why wouldn't B be moving away from A in the speed faster than light.  A simple explanation would be great.
37464
Could someone judge my (stoner) hypothesis that the speed of light has changed over time -- i.e. as the universe has expanded in volume light has slowed down, perhaps going so far as back to the big bang when it was infinitely fast and there was no time because everything happened at once etc. Thinking that the speed at which information can propagate through the universe is linked to the size of it seems intuitive to me. My question -- is there an easy disproof of this? Would Einstein have to be wrong? Does it violate anything supposedly more fundamental such as quantum or string theories? Do any current experiments invalidate it? If not can you show me in any case why you think its unlikely.   Edit 8/24  I'm accepting Mark M.'s answer but will post this here because there is a character limit on comments  @Mark M thanks, good answer but as someone who has only read some popular physics, and should leave this to the experts, I'm still muddled in my personal theory. I don't see why you should need two units to measure the speed of light. The thing I have a hard time wrapping my head around is the relation of time and distance. They seem like they could fundamentally be the same thing. If you say time is measured fundamentally by the vibration of so and so quantum object in space... why can't we just measure that vibration distance as the constant... I'll repeat myself to try to be clear... there is a certain minimum distance that particles have to go to interact with each other... if it wasn't vibrating there wouldn't be time,its what creates the illusion of time...so instead of talking about speed or c as distance/time.... can't we simply talk about that distance a quantum object vibrates....I'll lead up to my point.....perhaps there can need be only one constant here, and that is the physical size of the universe. a tiny metal tuning fork doesn't appear to be vibrating at all but if you blew it up to the size of the empire state building the metal rods would move from window to window. perhaps as our universe expanded in size the length of that minimum vibration (perhaps infinite at point zero) would have expanded, and therefore created the illusion of time and the speed of light, which, as the universe expands, will continue slowing down. perhaps we are like a big balloon and we have been blown up and all the fields/particles-without-size are vibrating more and more in that space.. am i missing something obvious here?
8314
To an an external observer it appears that time has stopped for photon. But this relation is reflexive, so for an observer travelling with the photon it appears the universe has stopped everywhere.  Is this right?  Space also gets distorted parallel to the direction of motion, but not perpendicular to it.   Does this mean that for an observer travelling with a photon sees spacetime as a flat plane?  Note 1: I'm using language vividly not literally when I say a photon experiences space and time. Not that I'm against idealist or panpyschist interpretations of matter or energy come to that.  Note 2: Since it's been pointed out that an inertial frame can't be boosted into the frame of a photon and moreover there is no limiting behaviour here - no matter what frame you are boosted into, light will always appear to travel at c. Nevertheless, when an external observer watches a train chasing a photon by gradually increasing its speed, he will note that the train speed is approaching the speed of the photon (but will never match it) and when he looks at the clock inside the train he will not too that it is gradually slowing down (but never actually stops). Using this picture, we can attempt the thought experiment - if not one that can be carried out - of what a world looks like to a photon.
99997
So, I just saw  by Veritasium that discusses how it's impossible to measure the one-way speed of light from a light source to a detector, since it's impossible to synchronize clocks in a fashion that would prevent a directional difference in the speed of light from altering clock speeds via relativistic time dilation in such a fashion that it would conceal directional differences in the speed of light. As a result, it's only possible to measure the two-way speed of light, where light passes from a source to a reflector and then back to a detector located at the light source, and it's impossible to rule out a directional difference in the speed of light, which could be as extreme as light traveling at c/2 in one direction, and instantaneously in the other. However, Veritasium goes on to briefly mention that if there is true, we would observe stars in the direction of the instantaneous speed of light in real time, rather than through a time delay of hundreds or thousands of years. During his papers on relativity, Einstein noted this possibility, and assumed that it traveled at equal speeds in all directions; as a result, this assumption has apparently been called the &quot;Einstein Synchronization Convention&quot;. This suggests to me that if the Einstein Synchronization Convention is untrue, and there is a directional difference in the speed of light, we should be able to notice this through astronomy; if the light takes less time to travel to us from some directions than others, we should see older stars and galaxies in the directions where light travels more quickly than in the directions where it travels more slowly, since we'll be observing them as being closer to &quot;real time&quot;. For instance, if you had galaxies 3 billion light years away from us, but it took light 2 billion years to reach us in one direction and 4 billion years to reach us in the other direction, you'd expect the galaxies in the 2 billion year direction to look 2 billion years older than the galaxies in the 4 billion year direction. Is this correct, or would a variation in the speed of light invalidate our methods of measuring distance and/or the age of celestial objects? Have there been any astronomical studies investigating this, and if so, what have they found? To be clear, I am specifically asking about detecting directional variations of the one-way speed of light using astronomy. I don’t want answers that discuss variations of the speed of light due to the velocity of their emitters (which was disproved by experiments prior to Einstein). I don’t want answers talking about clocks or simultaneity, since they’re not really directly relevant to this question.
78875
I have currently been introduced to relativity of simultaneity with the following thought experiment:  Consider a train in which there are two presidents, one sitting at the front and the other at the back of the train. The two presidents must sign an agreement simultaneously for everyone to be happy. The way that they are going to carry this out is by having a laser shoot a light beam simultaneously towards each of the presidents and when the light beam gets to each president they will sign the agreement. Now consider this train moving from left to right and we will analyse this situation from the two inertial reference frames "train frame" and "platform frame". It then follows that from the "platform frame" the two presidents have not signed the agreement simultaneously while regarding this situation from the "train frame" the presidents did sign the agreement simultaneity. Until this point I understand everything  Q: Now consider that there is a bomb in the middle of the train and that bomb would blow up if the two presidents don't sign the agreement simultaneously and won't blow up if they do (this can be achieved via sensors added at the back and front of the train). Now this experiment is carried and from " platform frame" train blew up while in "train frame" train didn't blow up. Now this poses a paradox as the train can't blow up and not blow up in the same time even if its a matter of reference frames, since reality is connected. How can this paradox be resolved?   Moreover from this paradox I get the feeling that, even though observers in different inertial reference frames don't agree on simultaneity, they must agree on something, maybe cause of their action on nature around them? I don't know how to call this agreement. Because if they wouldn't agree on this (idk how to put it into words) then we would get paradoxes as the one above. Thank you for any help  ps: The though experiment with the presidents was presented by Brian Greene in his course on special relativity
7159
There is a relativistic paradox I was introduced by a friend in Ithaca (Courtenay Footman). It involves a vehicle travelling over a ravine relativistically, so it's in the general family of ladder paradoxes. There are many ways to phrase it, but here's a simple one.   A three-car train with each car of length $L$ is travelling over a bridge across a ravine. (Assume that the couplings between cars are stiff: this train does not bend.) Unfortunately, some evil genius has precisely removed a length $6L$ of the track. The train however is relativistic, travelling at a speed such that $\gamma \approx 12$.  In the reference frame of the ground, the train should appear to be 1/24th the size of the gap, which suggests that it will fall a little and should crash into the bridge opposite. But in the reference frame of the train, each car of the train is at least twice as long as the gap and aside from the very front wheel and the very back wheel, which need to be supported by rigidity, everything is constantly well-supported. This suggests that the train makes it. Who is right?  I vaguely remember that as an undergraduate knowing only a little relativity, I trusted the train's perspective because it's at rest in that perspective, and Courtenay said that this was ultimately correct after a lot of details got resolved. But I do not remember all of these details such that I could work it out for myself. So I'm looking for some discussion that's a little more well-sourced or authoritative.  What is this paradox called, and what literature has addressed the issue?
35762
While it is fairly established that both fast movement and the presence of gravity make time pass slower as compared to a system at rest / free of gravity, does that mean that there is no way for time to pass faster than in vacuum, or does general relativity also have "faster" metrics?  To be more precise, is there any frame into which one could go for a while, and on returning to vacuum less time would have passed there?