Spacetime interval
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Although different observers may measure different time lapses and different distances between the same two events, there is an underlying quantity which remains the same -- the spacetime interval.
[edit] Formula
The spacetime interval is given by the formula
I2 = t2 − x2
or, if x is greater than t, you can use
I2 = x2 − t2
In this formula, I is the spacetime interval, t is the elapsed time between the two events measured in any one particular frame, and x is the distance between those two events in that same frame. Also, note that x and t must be measured in the same units since the formula calls for them to be subtracted from each other. This is accomplished by using the constant speed of light to convert units of time to units of length or vice-versa.
The only catch here is that the x-axis has to be parallel to the direction of relative motion between any two frames which are being compared.
[edit] Example
Two detectors are set up -- one at each end of a hallway that is 40 m long -- to detect flashes of light. A small robot is programmed to zoom down the hallway, emmiting a flash of light at the beginning and end of the hallway and recording the times that it emits those flashes. Here are the data collected by the robot and the detectors:
| Event | Robot coordinates (t,x) | Detector coordinates (t,x) |
|---|---|---|
| Flash at beginning of the hall | (0 m, 0 m) | (0 m, 0 m) |
| Flash at the end of the hall | (30 m, 0 m) | (50 m, 40 m) |
Do these data agree with each other? Well, at first glance, it seems that their clocks must not be working right because they measure different times between the same two events. But that is, in fact, what special relativity predicts. You can't just compare elapsed times, you have to check the spacetime interval.
For the robot: I2 = t2 − x2 = 302 − 02 − 900 so I = 30 m.
For the detectors: I2 = t2 − x2 = 502 − 402 = 2500 − 1600 = 900 so I = 30 m.
The spacetime interval between the two events is the same.
[edit] Math or Science?
The above example was created to make the numbers come out the way they did. There were no real measurements done. To see if the spacetime interval is a "real" thing, we would have to set up a situation like the example and make real measurements. How hard would that be? Well, to get an idea, you should investigate how fast the robot would be going in the above example and how much time would pass, in seconds, between the two events in each frame.
Speed of the robot relative to the hallway:
= 0.8 or 80% the speed of light (that is, around 240,000 km/s)
Elapsed time (in seconds) measured by the detectors:
= 0.0000001667 s
Clearly, this is a very hard experiment to pull off. You have to accelerate the robot to an amazing speed and be able to detect a time lapse that is very small. But we do have the ability to do this experiment with subatomic particles, and the results do support the claim that the spacetime interval remains the same while the measured time lapses and distances are different for the "robot" and the "detectors."
