guitarphysics said:
I couldn't really understand a lot of what Simon Bridge and mfb where talking about (as I said, I don't know much in the way of physics or math).
Don't worry, that was mostly just me trying to understand the question. Don't be afraid to post the context - it can be very helpful. Had you said "Brian Greene's book" or something earlier the replies would have been more understandable sooner.
I will second bcrowell's book suggestion. iirc it does not have the emphasis on the wierdness that most seem to.
What is your physics level?
I am trying to improve my ability to describe these things so I'll give it a go, and the others can suggest improvements, then we'll both learn something :)
I'll try and start with a secondary school senior level of physics, and no calculus... that will limit how detailed I can be and you should realize that it is unlikely to do justice to the subject. You will need to know about vectors and coordinates in normal 3D space.
I will be using math symbols a lot - try not to let them intimidate you ;) I'll try to define things as I go so the math will be a kind of short-hand.
A "normal" 3D position would be ##\vec{q}=(x,y,z)##
a 4D position would be ##\mathbf{q}=(ct,x,y,z)## we also write ##\mathbf{q}=(q_0,q_1,q_2,q_3)## all by convention.
You can just match up the same positions so ##q_0=ct,q_1=x,\cdots##
Notice that we use ct instead of just t for the "time" coordinate - this is to be consistent: the units of position have to be some length and you cannot mix units up in one vector.
To turn time into a length-like thingy, we have to multiply it by some speed (because "speed = distance over time" right?) In the past the speed we picked to use for this may have been the speed of the King's favorite racehorse on a fine day timed on the King's stopwatch. This would be problematic - for one thing, the King would have to do all our measurements and he's a busy man.
We pick the speed of light in a vacuum, instead, because it has the special property that everyone measures it to be the same no matter what and so everyone can agree what it is.
This fact that the speed of light in a vacuum is the same for everybody is the cornerstone of relativity. The consequences of special relativity follow from this. Because it is the same for everyone it is considered a fundamental property of the Universe.
The magnitude of ##\vec{q}## would be ##q=\sqrt{x^2+y^2+z^2}## - that's the rule for finding out the distance to a position. The rule follows from the geometry of Euclid (look him up) which most of us think of as "normal" geometry. It's basically Pythagoras.
You'd expect the rule for a 4-vector to be the same, only in 4D - like this: ##|\mathbf{q}| = \sqrt{q_0^2+q_1^2+q_2^2+q_3^2}## and this is the kind of rule that Brian Greene uses as a teaching guide in his book. But he's telling the general public one thing and he's telling his students another thing ... what he tells his students is this:
##|\mathbf{q}| = \sqrt{-q_0^2+q_1^2+q_2^2+q_3^2}## [1]
... spot the minus sign? This is what happens in
our Universe.
Since it obeys a different rule to Euclids, we tend to talk about these things being "in 4-space" rather than "in 4D" but it is usually clear by context which kind of 4D is intended.
If something zips by you in, say, the z direction, then you can write it's velocity, u, like this: ##\vec{u} = (0,0,u_z)## and ##u_z=\Delta z/\Delta t##.
In 4-space you'd have to write something like: ##\mathbf{u} = (u_0,0,0,u_3)## since the object is clearly moving in time as well as space - but is not moving in the x or y directions.
How do we work it out?
By analogy, it seems we'd want to do something like ##u_3 = du_3/dq_0## since ##q_0## is what we think of as the 4-space equivalent of the time-axis. But notice that the time we divided by in the 3D version was the time measured by someone who is not relativistic, and the speed has to be non-relativistic as well (or the classical picture would not work).
In relativity we call this the "proper time" and give it the symbol ##\tau##. Proper time is related to "regular" time by ##\Delta t=\gamma \Delta\tau## where the ##\gamma## is a factor that depends on the 3D relative speed. This is the "time dilation" effect you have heard about. [3]
So we want to define the 4-velocity more like: $$\mathbf{u} = \frac{\Delta \mathbf{q}}{\Delta\tau} = \left (\frac{\Delta q_0}{\Delta\tau},0,0, \frac{\Delta q_3}{\Delta\tau} \right )$$... and ##\tau## is the time as measured on a clock carried by the object.
But we know that ##\Delta q_0=c\Delta t## and ##\Delta t = \gamma \Delta\tau## ... so $$u_0=\frac{\Delta q_0}{\Delta\tau}=\frac{c\gamma\Delta\tau}{\Delta \tau}=\gamma c$$ ... if the speed were zero (object at rest) then ##\gamma=1##, ##u_3=0## so the 4-velocity will be ##\mathbf{u}=(c,0,0,0)## pretty much automatically.
Back to the moving object ... we also know that ##q_3=z## and using ##\Delta t = \gamma\Delta\tau \Rightarrow \Delta\tau = \Delta t/\gamma## we can write: $$u_3=\frac{\Delta z}{\Delta\tau}=\frac{\Delta z}{\Delta t/\gamma}=\gamma\frac{\Delta z}{\Delta t} = \gamma u_z$$... so, the object has a 4-velocity vector: ##\mathbf{u}=\gamma(c,0,0,u_z)##
All this just drops out automatically from the math.
The key to everything here is understanding ##\gamma## - which is given by: $$\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$$... in the above case, ##u=u_z##. This factor is a direct consequence of all observers measuring the same speed for light in a vacuum.
And that is a whole lecture by itself. [4]
(There are some bits in there that I'm not sure if I should have done a bit more, or a bit differently. No doubt someone will tell me.)
---- footnotes --------------------
[1] there is another rule that goes like this ##|\mathbf{q}| = \sqrt{q_0^2-(q_1^2+q_2^2+q_3^2)}## but they are equivalent.
[2] If we use a very small change, we write: ##u_z=\delta z/\delta t## and for an infinitesimally small change we get ##u_z=dz/dt## ... which is the calculus you see in places like
wikipedia.
[3] ##\gamma## is always bigger than 1, and approaches infinity as u approaches c. This is one of the consequences of c being the same for everybody. It means that if the guy on the spaceship (going fast wrt you) has a clock, and you see it's second hand click forward by 1 second, then your clock will tick off ##\gamma## seconds... so he always seems to be slow.
[4] further reading:
Relativity and FTL - despite it's title, is a fun and accessible introduction to the fundamentals of relativity for people who are a tad math declined.
