relativity – Physics Notes

Another Twins Answer

Why should the twin on the spaceship be younger than the other on earth if each of them is supposed to observe the time dilation of the other in his own frame?

The question of which is younger when they are apart and in relative motion has no answer unless we specify the observer who is making the comparison (which could be either of them – or perhaps some other arbiter such as one who is stationary with respect to the Cosmic Microwave Background radiation).

Once they reunite they, and everyone else, will agree that the one who ends up younger is the one who experienced more acceleration towards the other when they were far apart (or more precisely for whom the integral of distance times the negative of its second derivative is greatest). But even though they will agree on the end result, they won’t agree on a moment-by-moment accounting of how their ageing rates compared.

Source: (1000) Alan Cooper’s answer to Why should the twin on the spaceship be younger than the other on earth if each of them is supposed to observe the time dilation of the other in his own frame? – Quora

In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey?

The traveller’s clock can indeed be unambiguously compared with each space station clock at the event where they pass by one another, but that is still only comparing clocks when they are at the same location. And the problem with saying that comparing one’s time with that on a space station is equivalent to comparing it with the one on Earth is that it depends on agreeing that the space station clocks are properly synchronized. But if the space station clocks appear synchronized with the Earth clock in its own frame, then they will not appear synchronized to the traveller who is passing by them. So the time on the space station clock does not match the traveller’s idea of what is the current time back on Earth.
One can indeed go through the process of keeping track of the space-station clock times compared to the traveller’s clock, and will find that those recorded times are all greater on the space-station clocks by the same Lorentz gamma factor. But so long as the velocity remains constant, the traveller could be part of a lined up fleet of ships all moving at the same velocity past the Earth (and so stationary with respect to one another with the Earth and space stations moving past them), and if they all synchronize their clocks with the traveller then the Earth and space station clocks will record the intervals between successive ships of the fleet as greater than the time differences between the clocks on those ships. In other words the Earth (and space station) observers see the ship times as more closely spaced than their own and the traveller (and fleet ship) observers see the times on space station clocks as more closely spaced than the times (on their own ship-based clocks) at which they pass by them. At first sight perhaps this looks like a paradox, but we need to note that each observer of either kind is comparing times on different clocks of the other kind with successive times on the same clock of their own and each can attribute the effect to an assumption that the other set of clocks is not properly synchronized. So this isn’t really a paradox, but there is still no way of deciding which team is actually synchronized and which is not – and without being sure of that the traveller can’t rely on the space stations as true representatives of the time back on Earth.
Making the traveller turn around and return to Earth is just one way of getting some particular pair of clocks back together for an unambiguous comparison of time intervals. (Another would be to have the Earth chase after the traveller and compare notes when she catches up, and yet another would be to do things symmetrically.) But they all involve having someone change their inertial frame (ie accelerate) and the result depends on the acceleration pattern but is always basically that the one who experienced the most acceleration towards the other when they were far apart is the one who will end up younger.

[In the symmetrical twins story both end up the same age, and are not surprised because each has seen the other age first more slowly and then more rapidly but ending up with exactly the same total amount of ageing as they themselves have experienced. If they use the light travel time to infer when each tick of the other’s clock actually occurred (as opposed to when they see it), then each will infer that the other’s clock was running more slowly during both constant speed parts of the trip, but more rapidly during the period when they felt the force of acceleration during the turn-around process – with the same final result.]

Source: (1000) Alan Cooper’s answer to In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey? – Quora

Where in the universe can we find such an inertial frame? Certainly not on the surface of earth!

SR only applies exactly in the absence of gravity. So in the real world it is just an approximation that works well enough for predicting things where the effect of gravity is small (such as interactions between small high velocity particles in accelerators near the Earth’s surface, or between spacecraft and small bodies like asteroids far from planets, but not for things like apples falling out of trees on Earth).
In regions where it does provide a good approximation, it works just as well for accelerated as unaccelerated frames, but for accelerated frames the formulas needed to express physical laws in terms of the observer’s coordinates are more complicated.

Source: (1000) Alan Cooper’s answer to In twin paradox, the traveller’s clock ends up with a lesser total elapsed time, so we can tell who made the trip. Does this not contradict the postulate of SR that all physical laws are the same in all frames and all inertial frames are equivalent? – 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.

Ron Davis’s answer to Who clocked the speed of light and how did they clock it? What instrument did they use and when? – Quora

Source: (1001) Ron Davis’s answer to Who clocked the speed of light and how did they clock it? What instrument did they use and when? – Quora

Lorentz Expansion!

A Quora question asks:

Two guns on Earth D metres apart fires simultaneously at a metal plate moving by at close to light speed. Is the distance between the two bullet holes in the plate length contracted?

It depends on who thinks the guns fired simultaneously.

If, as seems most likely, the question means that those Earth-based guns were synchronized by someone on Earth with them, then the distance between the two holes, as measured by any observer stationary with respect to the Earth, is (and remains) exactly equal to the distance between the two guns. Since the plate appears contracted to these Earth-based observers, if it was marked with units of length in its own rest frame those markings would appear closer together to the Earth-based observers and so there would be more of them between the holes than the number of length units measured on Earth.

In other words the distance between the holes would appear to be

greater from the point of view of someone travelling with the plate.

It may seem puzzling that this happens despite the fact that from the traveller’s point of view the distance between the guns is “length contracted” and so appears to be less than that measured between them on Earth.

The puzzle is resolved by the fact that from the point of view of the traveller the guns did NOT fire simultaneously. The one making the front hole appears to the traveller to have gone off earlier. (And it is a worthwhile exercise for anyone seeking to learn about relativity to work through the calculation needed to show that the delay is by exactly the right amount for the forward movement of the plate to create the observed bigger distance between the holes).

Alternatively, if the traveller thinks the guns went off simultaneously, then the Earth-based observers think there is a delay. (And again, working out the details is a worthwhile exercise for any beginning student of the subject.)

P.S. The question of whether or not a length or object is “length contracted” does not really make sense without any mention of which observer is doing the measurement.

Source: (1000) Alan Cooper’s answer to Two guns on Earth D metres apart fires simultaneously at a metal plate moving by at close to light speed. Is the distance between the two bullet holes in the plate length contracted? – Quora

What does it mean to say that there is a distance between two events in time? (Another Quora Question)

It probably means that the speaker is taking a Galilean approach to physics.
In modern relativistic physics, the property of having a time-like separation between two events is independent of observer, but the magnitude of that separation depends on the observer. And two events which are spacelike separated, while having no time difference for some observers, will still appear to have a non-zero time difference for others.
So the concept of a time (or space) “distance” (ie a specific value of the difference) between two events in space-time does not make sense without reference to an observer.
However in the case of two time-like separated events the time difference is nonzero for all observers, and if we restrict to inertial observers it has a nonzero minimum (which corresponds to the time difference as seen by an observer who experiences both events directly without any intervening acceleration or gravitational field gradient). But although this minimum is in principle computable by any observer it does not correspond to the time difference actually “seen” by that observer.

Source: (1000) Alan Cooper’s answer to What does it mean to say that there is a distance between two events in time? – Quora

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!

Source: (1000) Alan Cooper’s answer to 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? – Quora

More TwinStuff from Quora

Source: (1000) Alan Cooper’s answer to In the twin paradox where does the missing time go? If the twin turns back to Earth then turns away again their notion of now switches back to the past. What does this mean for the experience of the observer on Earth relative to the moving twin? – Quora

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.