In this thread, Chronos said space (distance) and time can be measured in the same units. Does that mean both have the same units? Can someone explain this to me? Are they one in the same?
If you multiply a quantity with units of time with a constnat with units of speed then you get a quantity with units of distance. so ‘ct’ (the speed of light multuplied by ‘time’) has units of distance. In relativity as space and time become mixed is necessary to measure the two in the same units, thoguh usually for convienence natural units such that c = 1 are chosen.
I believe Chronos is referring to natural units.
For practical engineering purposes, and everyday life purposes, you’d probably prefer to keep time and distance as separate physical quantities, with different units — just as they’ve always been in classical physics.
Our usual units for time are totally arbitrary. What is one second? But the distance light can travel in a given amount of time is not arbitrary. Therefore distance is in some sense the best unit to use for keeping track of time. Does this mean that in a super-deep sense, the units of space and time are really the same? No, but they are deeply connected.
Classical units of distance were just as arbitrary as the units of time.
In the modern SI system it’s actually the second that’s defined “first” — as so-and-so many periods of a particular emission of radiation — and the meter that’s defined indirectly, in terms of c and the length of the second.
My desk is 5 nanoseconds wide, and I’ve been sitting at it for 180 gigametres.
55 nanoseconds per fortnight - it’s not just a good idea, it’s the law.
Right. And civil engineering practices don’t really take notice of relativity at all, afaik. The tolerance of the materials you use to build another bridge across the East River is a lot worse than the amount of error that is introduced by using Newtonian Mechanics (which is a flat-out FALSE model of the universe) rather than Relativity. Newtonian physics is “good enough” and is a lot simpler. Now, that starship you were always hoping to build may be a different matter…
My car gets 40 rods to the hogshead and that’s the way I likes it!
You are completely correct: 1 second is just as arbitrary as 1 meter. I guess it would be more correct to say that there is a non-arbitrary way of equating 1 second with 1 meter: light travels 3*10^8 meters per second. If you define a meter in terms of some markings on a metal rod, then you can define the unit of time in terms of some number of rod lengths traversed by light. If you define time in terms of a periodic natural phenomenon, such as a cesium atomic clock (in the olden days it was lunar cycles, seasons, pendulums, etc), then you can define the unit of distance in terms of how far light travels during a certain number of periods. In SI units, it happens that we chose the latter. I guess the point is that you can do it either way – the units of time and distance can be thought of as the same except for a factor of c.
Draw 2 perpendicular axis. Label the vertical axis “time” and make its unit of measure years.
Label the horizontal axis “distance” and make its unit of measure also years (light years).
A photon’s worldline would trace a 45 degree angle and its slope or velocity or “c” would equal 1 year/1 year =1.
Quoth iamnotbatman:
Well, physics can’t address the question of what’s “really true in a super-deep sense”, but a model of the world where they are the same units, and c really is 1, is a lot simpler than a model of the world which insists on separating them.
And back when I was taking relativity classes, we would keep track of how long the professor went over the allotted time in AUs. Usually, he ran about 2 AUs over.
My understanding (could be wrong) is that everything moves through 4D spacetime at speed c. Always.
So in 3D space, c is an upper speed limit and in 4D spacetime, it is an upper an lower limit.
What this means is that something that is not moving in 3D space, is doing all its moving in the time direction of 4D spacetime. Similarly, if something is moving at speed c in 3D space, it is not moving at all in the time direction in 4D spacetime.
So when looking at things in 4D spacetime, the unit that measures magnitude would have both a time component and a distance component but the two are really the same thing as everythng is moving at c anyay.
I welcome any corrections to the above.
That’s the way I see it. It can sort of make sense of that common, stupid rubber mat analogy for a gravitational field*. How can curved spacetime cause gravity if you are stationary WRT your massive object? It can because you’re not stationary in spacetime you’re falling through time at c.
- you have to flatten the mat out and add a third dimension to represent time but hey, it’s a stupid place to start.
It’s not really the right way to think of things as the “speed of something moving through spacetime” doesn’t really make sense, but at the same time it’s an interesting way of introducing spacetime.
If you think about it velcoity in non-relativstic physics is dx/dt, but in 4-D spacetime t is a component of x. You can still however define somethign called four velocity which is U = dx/dτ where τ is proper time. However you shouldn’t confuse U with being the exact analog of 3-velcoity.
The magnitude of the four velocity of any object that is not moving at c is equal to 1 (in natural units where c =1). This is as the four velocity is the unit (i.e. of magnitude 1) vector tangent to the object’s worldine. For objects moving at c all tangent vecotrs are null (i.e. have a magnitude of 0), so the four velcoity of an object moving a c is undefined.
Nitpick: One and the same
An important note on this is that “magnitude of a vector” does not mean the same thing as you’d normally think of when dealing classical vectors. For a (special) relativistic 4-vector, (t, x, y, z) in natural units, the magnitude is sqrt(t^2-x^2-y^2-z^2), rather than the sqrt(t^2+x^2+y^2+z^2) you might expect.
That is how the tangent vector for an object moving at c can have magnitude zero despite not being zero itself.
Let me guess, leahcim– You’re a particle physicist, right? Relativists usually use the other sign convention, with plus signs on the space parts and minus on the time.
It’s been a while, but back in the day…
Isn’t there an alternate convention where you always multiply the time component (in a four-vector) by i, so that when you square it in the expression for vector magnitude, a -1 gets baked in automatically?
That’s “automatically”, if you don’t consider including the i everywhere a hassle — which is how these tricks seem to me when I read about them.