special relativity / metric equation

[theMan]

i'm taking a class on special relativity right now, and there's one thing that's really bugging me.

Say you start from rest (event A) and accelerate to a relativistic speed and then crash into a wall (event B). your watch will measure the proper time interval between A and B. If you're in an inertial reference frame with synchronized clocks at A and B, an outside observer will be able to measure the coordinate time between the two events by looking at each clock when each event occurs there.

then, using the metric equation, (?s^2 = ?t^2 - ?d^2) you can calculate the spacetime interval between the two events. Ok... but what does that even mean? The spacetime interval by definition involves a clock moving between two events with no acceleration. in this case, my clock is accelerating, and therefore cannot measure the spacetime interval. how can the spacetime interval exist if it isn't being measured? am i supposed to just imagine some thing moving through spacetime that doesn't actually exist? it just seems like a meaningless quantity...

maybe my question is just overly philosophical and I should just be a good student and calculate the numbers, but something about this seems strange.

I can accept that you can take two arbitrary events plotted in spacetime and draw lines between them to create different types of time intervals, but if there's actually a real object moving in spacetime, don't you just have to accept that you've measured whatever type of time interval you've measured?

oh, and let me know if this entire post made any sense, becuase i'm tirrrred as balls right now.

 

[iCyborg]

I'm not an expert on this, but no one else answerred so...

I am a bit confused by your post. Are A and B events or locations, you seem to use both? If locations, I'll assume synchronizing clock also means A and B are at rest relative to each other.
I'm not sure about this definition of spacetime interval, but if you have two points A and B in spacetime, then if you have acceleration, the path from event A to event B will be curved, and to compute the proper time experienced by the traveller, you have to do an integral along the curve of that ds that you gave above (notice that it's just the length of the curve under that metric). If there's no acceleration, then the path is a straight line, and you can just take the interval to be the measured time (you'll get the same answer if you integrated along the line). This particular metric has a "weird" property that a straight line is the "longest" path between two points. Hope this make sense as I'm not really sure what bothers you.

 

[theMan]

Ok, A and B are fixed events in spacetime. In a certain situation, I am accelerating between A and B. Additionally, this whole situation is occurring and an inertial reference frame. So if you draw it out, my path would a curved line. An observer fixed in the IRF measuring the time it takes for me to get from A to B would be able to use basic kinematic equations to determine the coordinate time of the interval from A to B. My watch would measure the proper time from A to B.

NOW.... there's a special case of proper time called the spacetime interval, which is just a straight line between A and B (no acceleration). If asked to calculate what this interval would have been, had I moved from A to B without acceleration, I could -- using the metric equation (?s^2 = ?t^2 - ?d^2 where ?s is spacetime, ?t is coordinate time, ?d is spatial distance) . But since I'm NOT moving in a straight line, the spacetime interval was not measured in the problem, so how does it even exist?

to summarize, I'm given a problem where an object is accelerating through spacetime. then I'm asked to calculate the spacetime interval. I don't understand how a spacetime interval was involved in the problem (since its accelerating), and therefore it makes no sense to calculate it. are you just supposed to imagine some imaginary watch moving through spacetime between the two events, even though this watch doesn't actually exist or was not even mentioned in the problem???

thanks for attempting to decipher my previous post btw.

 

[iCyborg]

I think my previous post answers this, so I can only try to clarify:

Let's take an analogy in R^2, say A=(0,0), B=(1,1), and let's use the normal metric (i.e. + +). Say we have two paths, one is a straight line, the other along y=x^2 and they both start at A and end B.
The length of the path in the straight line case is just sqrt(?y^2 + ?x^2) where ?x=?y=1 for our case.
In the other case the length is given by Integral(sqrt(1+4*x^2)dx) from 0 to 1 (it's a special case of Integral(sqrt(1+f'(x)^2)dx) where f(x)=x^2, see http://en.wikipedia.org/wiki/Arc_length#Modern_methods).
Notice that this also covers the straight line case: you could integrate the same expression for the function f(x)=x, and you would get exactly the same answer sqrt(?y^2 + ?x^2).


Now in the spacetime case, the metric is (+ - - -). The key thing is that the length of the curve in this metric represents the proper time measured by someone travelling along that curve.

If someone is going from A to B with no acceleration, then it's the same case as above for the straight line, and the proper time is just ?s = sqrt(?t^2 - ?d^2), i.e. what you have said.

If you accelerate, your path will be curved, so to find the length of the curve, you need to sum up all the little infinitesimal ds (not ?s now) along the curve, in other words you need an integral of ds along the path from A to B. On each of these little pieces of the curve, you can use the same formula that the elapsed time is the spacetime interval (I assume by that you mean ?s). Notice here that you could also integrate little pieces of the straight line, but again, it's just ?s between final endpoints A and B.

Let's say you have an observer at rest relative to A and B (and clocks at A and B measure the same of course). Then whatever path someone took from A to B, his measured proper time will be exactly the coordinate time, since ?d=0 for him.

Perhaps this is what's confusing you: we don't have a single spacetime interval, but we can collect all the infinitesimal ones and sum them up via integration along the curved path to get his proper time.
Because, everything here exists and is properly defined:
1) the observer at rest relative to A&B will measure the coordinate time.
2) the one going from A to B in a straight line will measure a little less, since ?d^2>0 means that ?s^2 = ?t^2 - ?d^2 < ?t^2 i.e. the coordinate time
3) the observer who accelerates from A to B will measure whatever the integral gives him/her, and it can be proven that it's always a little less than what the observer in 2) measures.

If you go back to my first post, I'm actually trying to say the same thing, albeit in a somewhat terse and unclear way

 

[theMan]

Ok, I think my question is more philosphical than mathematical. I definitely understand everything you have said. My qualm is with your statement #2:

Perhaps this is what's confusing you: we don't have a single spacetime interval, but we can collect all the infinitesimal ones and sum them up via integration along the curved path to get his proper time.
Because, everything here exists and is properly defined:
1) the observer at rest relative to A&B will measure the coordinate time.
2) the one going from A to B in a straight line will measure a little less, since ?d^2>0 means that ?s^2 = ?t^2 - ?d^2 < ?t^2 i.e. the coordinate time
3) the observer who accelerates from A to B will measure whatever the integral gives him/her, and it can be proven that it's always a little less than what the observer in 2) measures.

In the problem I was given, there was a realistic situation. A particle was being accelerated in a circular path by a magnetic field. Then, it asked me to find the spacetime interval between A (where it started) and B (where the particle crashed into something). There WAS NO OBJECT MOVING IN A STRAIGHT, NON ACCELERATING PATH. In statement #2, you say "the one going in a straight line". In the problem, there WASN'T an object going in a straight line. Am I supposed to create an imaginary path? If so, what is the significance of this interval? It seems like an arbitrary quantity that has nothing to do with the actual situation in the problem.


do you see what i'm getting at? it's sort of one of those "if a tree falls in the woods and nobody's around, does it make a sound?" kind of things. If there's nothing moving in a straight path, why does it make sense to measure what the time interval would have been had something been moving in a straight path?

 

[iCyborg]

I see. Well, a spacetime interval is a term on its own and just measures the distance between A and B in spacetime, and it happens to be the same as the proper time someone moving on a straight line would experience (of course, it doesn't just happen, the metric was conveniently chosen, but you get what I mean).

I think it's similar to having a particle move on a curved path from A to B in normal 3D space, and then asking what's the distance between A and B. You could also say that this distance between locations A and B is something some imaginary odometer would measure if it went in a straight line. Do you think this distance is meaningless because in this situation too, no object was actually moving in a straight line?

E.g. in spacetime if you wanted to go back from B to A (not exactly as spacetime events, unless you have a time machine), you might find the spacetime interval information more directly useful than whatever you measured on your curved path. In any case, I think it makes sense to ask for this in a class example or homework or something.