Yes. Any decent introductory text on relativity does this – but probably just in one or two sentences before going on to derive the actual formula (for which the apparent level of complexity of the resulting equations may depend on the reader’s experience).

The basic idea is that if two identical side-by-side trains are passing by one another and a light signal is sent when the ends from which it is sent are together, then if the trains are in relative motion the signal will reach the far end of one before the other. So if observers on both trains measure the same speed of light then their units of length and/or time must be different. Working out exactly how the coordinates used by each observer are related to those of the other does involve the use of mathematical formulas and equations, but they are well within the scope of high school algebra so whether or not you call them “complex” is a matter of perspective.

# Tag: time dilation

## Time Contraction

Special Relativity tells us that two inertial observers in relative motion each perceive the other to be ageing more slowly – ie each infers that the tick intervals of the moving clock appear to be dilated. But can time contract as well as dilate?

Yes, but with the proviso that the dilation or contraction is just a description of how the progress of one clock appears relative to another and that two observers will not necessarily agree on which events in their lives are simultaneous – and so can only compare average (rather than instantaneous) clock rates using the total time intervals on their clocks between events where they are together.

Two observers who separate and reunite will agree that the total time experienced by the one that felt more forces of acceleration (or of resistance to gravity if spending time near a massive object) will be less than that experienced by the other. This means that from the point of view of the one who was more accelerated (or spent more time at the bottom of a potential well) the clock of the other appears on average to have been speeded up (ie tick intervals appear contracted), while the one who remains unaccelerated interprets this as meaning that that the other’s clock tic intervals were, on average, dilated.

Source: *(1000) Alan Cooper’s answer to Can time contract, as opposed to dilation? – Quora*

## So the excuse used NOT to apply relativity theory in the twin paradox is a brief period of zero seconds at the turnaround point?

No one who knows what they are talking about has suggested “NOT to apply relativity theory”. On the contrary, the correct application of relativity theory leads to the conclusion that when the twins re-unite they agree on the fact that they have both seen the traveller age less. They just disagree on when during the trip the Earth-based twin aged faster. The one on Earth thinks it happened at a steady rate throughout the trip and the traveller (after actually seeing it during the return trip) thinks (after making the light travel time correction) that it happened quickly during the turn-around.

Prior to the turn around, each sees the other ageing more slowly (due to the Doppler effect) and, even after making the light travel time correction, *thinks* that part of that slowdown remains unexplained (and so in some sense is “really” happening).

But any claim that during the outbound journey “we know for a fact that the travelling twin is younger than the earth twin” (or vice versa) is completely false. There is nothing that is absolutely true about the relative ages of the twins until they are at rest with respect to one another.

## Given that the Lorentz transformation is symmetrical with respect to interchange of space and time, how does it lead to length contraction but time dilation?

This is a question that I am surprised to not have seen before (especially since I have had to remind myself of the answer more than once – including, I suspect but can’t be sure, from way back before I entered my dotage).

It is true that in one space dimension the transformation equations

[math]x’=\gamma(x-\beta t)[/math] and [math]t’=\gamma(t-\beta x)[/math]are completely symmetrical with respect to interchange of [math]x[/math] with [math]t[/math] and [math]x’[/math] with [math]t’[/math].

(and in the case of three space dimensions the same applies if [math]x[/math] and [math]x’[/math] are the coordinates in the same direction as the relative velocity, so it’s not got anything to do with the dimension).

So what is the difference?

Well here it is in a nutshell.

When we measure the length of a moving measuring rod, we look at both ends at the same time and so are looking at the spatial distance between two events at the same time in our frame of reference.

But when we measure the time between two ticks of a moving clock we are looking at the time difference between two events that are NOT at the same spatial position in our frame.

So the nature of the two measurements is not symmetrical with respect to interchange of space and time.

I may add some more explanation and diagrams to show how this does lead to contraction for the rod length and dilation for the tick interval, but I wanted to get this off my chest right away – and also to address a couple of natural follow-up questions.

Namely, what kind of measurements would give the symmetrical outcome? Are there situations in which these others might be relevant? And why do we instinctively prefer the ones we do?

So, for example, what kind of time measurement would be symmetrical compared to our usual rod length measurement (and so would give a “time contraction” rather than the usual time dilation)?

Since the rod length involves looking at both ends at the same time in our frame, the corresponding time measurement would involve looking at the interval between two ticks at the same place. But how can we do this if the clock is moving? Well we could if the clock was extended in space, and if we have a long train of clocks that are synchronized in their own frame, then you can easily check that observers who look at the time between the ticks right in front of them will actually see a shorter interval than that measured by the travelling system – ie a time contraction.

And going the other way, what kind of measurement would give a length dilation? Well that would have to be the symmetric version of our usual clock measurement. And corresponding to our usual measurement of the time interval between two ticks at the same place in the moving clock’s frame, interchanging space and time would have us measuring the spatial distance between events where the two ends of the rod are at the same time in the rod’s frame. For example the managers of the rod might set off flares at both ends in a way that they, travelling with the rod, perceive as simultaneous. If we measure the distance between where we see those two flares then it will indeed appear dilated relative to the length of the rod in its own frame.

So now we come to the final question. Is there anything really “wrong” about these alternative kinds of measurement? If so what is it? Or is there just something about us which makes us think of what we do as natural and the alternative as somehow, if not actually wrong, then at least rather odd?

Here’s what I think (at least for now). The thing that makes us prefer to measure lengths in terms of events at the same time in our frame but times in terms of events at the same place in the moving frame is the fact that we, as blobs of space time, are much more extended in time than in space. (This is evident in the fact that we live for many years but do not extend for many light years in our spatial extent – or equivalently that in units adapted to our own spatial and temporal extent the numerical value of c is very large.)

So here’s a follow-up question. Could we imagine an entity which was the other way around? (ie of brief duration but of great spatial extent) And from the point of view of such an entity would it make sense to define measurements differently (as suggested above to achieve the effect of time contraction and length dilation)?

OR is it more just a matter of causality?

P.S. This is a question and answer that I have been meaning to post for some time, but was prompted to do so by Domino Valdano’s excellent answer to another question (in which she covers pretty much the same ground with a slightly different way of expressing the ultimate reason for why we measure as we do – which I may yet end up deciding that I prefer to my own). Please do read that one too!

## Spacetime diagrams from POV of both twins in the finite acceleration versions of the twin “paradox”.

If properly worded *this* would have been a good question. From the comments attached to the question we see that the questioner is really asking for *two* diagrams, one showing the point of view of *each* of the twins rather than a single diagram showing the coordinate systems of both. And by the ambiguous condition of “constant acceleration” he means constant acceleration as perceived by the stationary observer rather than constant proper acceleration as felt by the traveler.

Of course the case of constant proper acceleration would be more realistic in the sense that it just requires the traveler to experience a constant g-force, whereas constant observed acceleration requires an increasing applied force (which would actually become unbounded as the speed got closer and closer to c). But for a limited time it is possible to keep adjusting the applied force so as to create a constant acceleration relative to the Earth’s frame and in that case the relevant part of the world line (in any inertial frame) is a simple parabolic segment (rather than the hyperbolic segment that would correspond to constant *proper* acceleration).

With the assumption of constant accelerations in the stay-at-home inertial frame, the spacetime diagram in terms of stay-at-home coordinates is just this:

Here we have a parabolic segment taking the traveler from the start event to where he reaches a cruising speed of , followed by a straight line segment or the bulk of the trip, then a parabolic segment for deceleration, a vertical segment for time spent at the destination, another parabolic segment for acceleration back towards home, straight line for the cruise, and the final parabolic deceleration phase.

In this diagram the coordinates are [math]t_H[/math] for the time on the stay-at-home clock and [math]x_H[/math] for the position in the stay-at-home coordinate system, and we will use the name [math]x^{I}_{HT}[/math] for the function which gives the traveler’s position in stay-at-home coordinates in terms of the time [math]t_H[/math] that the stay-at-home observer perceives as concurrent with the traveler’s arrival at that position. (The superscript I on the function name is to indicate that this is what he infers rather than what he actually sees). So the graph of [math]x_H=x^{I}_{HT}(t_H)[/math] shows what the stay-at-home thinks is the position of the traveler when his (stay-at-home) clock shows time [math]t_H[/math]. This is one interpretation of the homie’s “point of view” but it is not what he actually *sees*.

What the homie actually *sees* is delayed by the light travel time from the traveler (just as what we see of a distant star many light years away is not what it is actually happening there now but what happened that many light years ago).

So to get the graph of what the homie actually sees we must look at the point on the previous graph that is the source of a light signal reaching home at time [math]t_H[/math].

We can get a graph of what the homie actually sees by tracing down each light-line from the [math]t_H[/math] axis to where it meets the [math]x_H=x^{I}_{HT}(t_H)[/math] graph and plotting the [math]x_H[/math] value of that event as [math]x^{O}_{HT}(t_{H})[/math] (with the superscript [math]O[/math] identifying the position actually observed at time [math]t_{H}[/math] rather than that which was inferred to be simultaneous).

Now let’s look at things from the point of view of the traveler.

The vertical axis now corresponds to the traveler’s clock time [math]t_T[/math] and the horizontal lines either to distances that we want to associate with that time. If we want to plot what is actually seen by the traveler then for each [math]t_T[/math] we plot the position coordinate corresponding to the distance from which the signal is coming (as determined, eg, by parralax). and if we want to plot where the traveller infers that the homie actually is at the time [math]t_T[/math] we attribute the distance of the source seen at time [math]t_T[/math] to the earlier time [math]t_T-\frac{|x_T|}{c}[/math]

What the traveler actually *sees* at any event on his worldline is exactly the same as what is seen by an inertial traveler whose world line passes through that event with zero relative velocity (ie for which the worldline is tangent to that of the traveler at that event). Such a tangential traveler sees the values of [math]x^O_{TTE}[/math] and [math]t^O_{TTE}[/math] corresponding to a time [math]x^O_{TTE}/c[/math] earlier in his own frame – so that [math]x^O_{TTE}=[/math] and [math]t^O_{TTE}[/math]

events that are seen by him at the time his clock shows time [math]t_T[/math] with position along that line corresponding to the distance he measures (eg by parallax) to that event; or, in the case of the inferred view those that are inferred to be happening simultaneously with that [math]t_T[/math] click of the clock or to those distance [math]x_T[/math] which he measures (eg by parallax) to whatever event we are talking about.

When the traveler’s clock reads time [math]t_T[/math] he is at the event for which [math]t_H[/math] is such that [math]\int_{0}^{t_H}\frac{1}{1-v(t)^2}dt=t_T[/math] where [math]v(t)=at,v_f,v_f-a(t-t_f),0,-a(t-t_r),-v_f,-v_f+a(t_h-t)…[/math]

To find [math]x^{I}_{TH}(t_T)[/math] we have to use the [math]t_T=[/math]constant simultaneity space for the traveler and find its intersection with the [math]x_H=0[/math] worldline of the homie

The light signal that the traveler is receiving from homie at this event can be seen from the above diagram to come from [math]t_H=[/math] and we have [math]x_H=[/math]

And

## Is special relativistic time dilation a real effect or just an illusion? Given two inertial frames each observer finds that the clock of the other runs slower than that observer’s own clock. So who is right?

*This* is a pretty good answer except that I wouldn’t say either of them is right if they think that their perception of relative slowness represents something that is objectively true for all observers.

Time dilation is a real effect on the perceptions of observers (with regard to the rates at which one another’s clocks are ticking). Neither of them is “right” if they think there is any real sense in which the other’s clock is objectively slower. But neither of them is wrong about how it appears to them, so it’s not really an illusion any more than the fact that if they are looking at one another then their ideas of the “forward” direction are opposite to one another. What turns out to be more of an illusion is the sense we all have that there is some absolute standard of time which determines which of two spatially separated events occurs before the other.