I've been trying to learn about the speed of light and time dilation, but I'm at an impasse.
The presented facts say that if I travel around the solar system at 50% the speed of light and then come back to earth I will have experienced less local passage of time than them. I will effectively have traveled to their future. I've also read that gravity causes time dilation too.
I understand that space-time is a sort of unified thing and that affecting space affects time. It makes sense then that gravitational forces bending space will also bend time. But velocity? I can't wrap my head around it and I can't find a good explanation for it.
People cite orbiting craft and planes as proof of time dilation, since their clocks will go slower than those in the surface (and vice-versa). This certainly explains gravitational time-dilation, but not necessarily velocity. Can't the change in local time passage be caused solely by the gravitational bending of space-time?
Trying to find an answer, I came to a very recurring and frustrating example in texts that seek to explain time dilation. A man on a moving train throws a ball forward. Since he's moving with the train and the train is his point of reference, the ball to him only moves at the speed he threw it. But to a woman on the station the ball is moving at the speed of the train plus the speed it was thrown with. To some authors, this seems to open the mind to time-dilation understanding. To me it only explains the relative nature of motion. It says nothing of time.
Another example I've found: if person A speeds away from person B very quickly, A's clock will seem to advance slower from B's point of view. How is this time dilation, though? The difference can be explained by the longer time it takes light to get to B, can't it?
I realize I can't be right against the fine physicists out there, so I was hoping someone here could enlighten me. Where does the notion that velocity causes time-dilation come from?
Best Answer
Special relativity (let's leave aside GR for now) is notoriously unintuitive - generations of physics students have found this to their cost, so you are far from alone. So there is no simple intuitively clear explanation over what is going one. That said, I will attempt a quasi-intuitive explanation.
I think the mistake students make is to take time dilation in isolation. It's easy to think here's a phenomenon called time dilation: what causes it? What actually happens is that different observers will disagree about what constitutes space and what constitutes time and time dilation is just part of a bigger phenomenon.
As I sit here at my keyboard I'm not moving in space, but I am moving in time. So for some activity (e.g. from the start to the end of me typing this sentence) my $\Delta x = 0$ and my $\Delta t = T$ for some time $T$. However the bug eyed alien that has just zoomed past at $0.9c$ disagrees. The alien, seated at their own typewriter, sees me moving at $-0.9c$, so in between starting and finishing my sentence the alien sees that I have moved some distance $\Delta x = d$. But in SR space and time are linked, so if the alien measures a different $\Delta x$ they must also measure a different $\Delta t$. The two are linked by the relationship:
$$ c^2 \Delta t^2 - \Delta x^2 = c^2 \Delta t'^2 - \Delta x'^2 $$
where the unprimed $t$ and $x$ are what I measure and the primed $t'$ and $x'$ are what the alien measures. Even without doing the maths it should be obvious that because $\Delta x < \Delta x'$ it follows that $\Delta t < \Delta t'$. In other words when the alien times how long it takes me to type the sentence they measure a longer time than I do. For the alien my time has been dilated.
You've probably also heard of length contraction. Well this is the other side of the coin. Time dilation and length contraction always occur together because in effect some of the time is being converted into length and vice versa.
All this follows from a fundamntal symmetry called Lorentz invariance. This states that if we measure a property called the line interval and defined by:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
then the quantity $ds^2$ is an invariant and all observers will measure the same value for it. To get the equation above linking my $(t, x)$ with the alien's $(t', x')$ I just exploited this invariance to require that $ds^2 = ds'^2$.
All very well, but I have really only pushed the non-intuitiveness one stage farther down, since my explanation assumes Lorentz invariance and this in turn is unintuitive. Still, hopefully this allows you to get some handle on what is going on.