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
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?
“In the twin paradox where does the missing time go?” I am not aware of any “missing” time. One twin experiences less time than the other but there is no gap where any time goes missing. (There is however an apparent speed-up of the Earth clock from the point of view of the traveller while turning around, and in the physically impossible case of an instantaneous turn-around that would look like a gap in the traveller’s understanding of what was happening “at the same time” on Earth; but in any possible actual scenario it would just be an apparent speed-up rather than a jump.)
“If the twin turns back to Earth then turns away again their notion of now switches back to the past. What does this mean…” Indeed! Does that sentence actually have any meaning at all?
Perhaps what that second sentence is referring to is the traveller’s idea of the time that is “now” back on Earth. It is true that when we accelerate away from something we infer a slowing down of its clock at a rate proportional to the distance. If the distance is great enough this effect can make the clock “behind” us appear to stop, but beyond that distance (called the “Rindler horizon”), rather than see it run backwards we actually just don’t see it at all. (And, yes, the Rindler horizon perceived by an accelerated observer is indeed related to the Event horizon surrounding a gravitational singularity.)
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.