The man in this video says that if I was sitting still and someone was jogging by me, due to relativity, we would see Andromeda as it is days apart in time. This feels wrong to me, and I did degree level physics and relativity but I am happy to find out a new nuance in understanding if he is right.
Please explain it to me (or debunk him). If a big supernova happens, and I catch it on its first/last day, can I make it go away by jogging?
The video has given you an incorrect impression. You and the person jogging past you are seeing exactly the same light from Andromeda, so you will both see Andromeda in the same state.
If you and the jogger are asked to say what time is it ‘now’ on Andromeda, you will give different answers, but that is because you are in different reference frames, so an event on andromeda will have a different time coordinate in your frame than it does in the frame of the jogger.
Let's just do the LT, in the Standard Formulation:
Alice, in $S$, is "at-rest" here. Bob, in $S'$, is also here, but has a small non-relativistic velocity $v$ towards Andromeda.
The LT convert's Alice's un-primed coordinates to Bob's primed coordinates:
$$ t'=\gamma(t-vx) $$ $$ x' = \gamma(x-vt) $$
With $v$ on the order of 1 m/s, we get:
$$ \gamma - 1 = \frac{1-\sqrt{1-v^2}}{\sqrt{1-v^2}} $$
$$ \gamma -1 \approx \frac{\frac 1 2 v^2}{1-\frac 1 2 v^2}=\frac 1 2 v^2$$
so with
$$ \gamma \approx 1 + 5 \times 10^{-18} $$
we can set it to one, and the LT is:
$$ t'= t-vx $$ $$ x' = x-vt $$
The point here is that the LT is linear, as in $y=mx+b$: slope and intercept.
Special Relativity students focus always on the slope, $m$, which is time-dilation, but it is the intercept (The relativity of simultaneity) that causes the paradoxes. With $\gamma=1$, we're just dealing with the latter.
So the Andromeda galaxy has a world line:
$$ A(t) = (t, L) $$
where $L$ is 2.5 million light years.
If you ask Alice, "where is the Andromeda galaxy right now?", she will say:
$$ A(0) = (0, L) $$
Let's put that in Bob's coordinates:
$$ x' = (L-0v) = L $$
(which is in error by 130 km bc/ we ignored length contraction), meanwhile:
$$ t' = (0 - vL) \approx -3\,{\rm days} $$
So Alice's "now" at Andromeda is 3 days ago at Andromeda for Bob.
If there is a supernova in Andromeda at $t=2\,$days for Alice, it is in her future, but for Bob, it has already happened.
Of course, neither Alice nor Bob can know of this supernova because it is outside their light-cones.
If the supernova occurred (2,500,000 years - 1 day) ago, then both Alice and Bob will physically see it tomorrow, and it will have happened. Alice puts this supernova at:
$$ E = (-2,500,000\,{\rm years} + 1\, {\rm day}, 2,500,000 ly) $$ $$ E = (-2,499,999\frac {364} {365}y, 2,500,000 ly) $$
We can introduce a 3rd frame, Charlie ($S''$), moving ultra-relativistically at $\beta$, away from Andromeda.
His time coordinate for the SN are:
$$ E''_t = \gamma(E_t + \beta E_x) $$
which is negative if
$$ |\beta| > \frac{E_t}{E_x} = \frac{2499999\frac{364}{365}}{25000000}$$
That means, for Charlie, the supernova hasn't happened yet, even though Alice and Bob say it happened millions of years ago, and its light is closer to Earth than Voyager 2.
So, as seen at https://en.wikipedia.org/wiki/Rietdijk–Putnam_argument:
That no inherent meaning can be assigned to the simultaneity of distant events is the single most important lesson to be learned from relativity.
— David Mermin, It’s About Time
The classical idea of time is that the universe exists at a single instant, and that instant is now. The universe keeps changing. The past is the state the universe used to be in. It is over and gone. The future is the state the universe will be in. It hasn't happened yet. The Newtonian universe is deterministic, so you could figure out the future before it happens. The quantum universe is not, so there are limits on what you can figure out. This is Presentism.
The hardest part of relativity is letting go of the idea that the entire universe exists in the now instant.
Space time diagrams help. If nothing else, they illustrate how that idea doesn't work. Two different observers see different slices of space time as the now instant. This leads to the idea of the Block universe. See What is time, does it flow, and if so what defines its direction?
This is the idea that other times exist, even though you aren't using them now. It is like the idea that other points in space exist, even though you aren't using them now. This is Eternalism.
It seems that Eternalism is forced on you by relativity. But there is another way of thinking about it that I find more satisfactory. You exist in the now instant. Your past is done and your future hasn't happened yet. You trace out a world line. The Andromeda galaxy also traces out it's world line. You both have a now instant at every point of your world lines. but there is no universal way of matching them up.
The difference between this and the Block Universe is purely philosophical. You use the same space time diagram to reach the same conclusions. You still have conceptual hurdles to wrap your head around. I like it better, but many people prefer the Block Universe.
