Relativity & Spacetime: Interval Explained

[cuchey615]

Hi! I'm a student reading a book given to me by my teacher about relativity and spacetime. It says that the separation between events in spacetime is measured in "intervals," and I can understand that part.

What I don't understand is why you subtract the squares of the distance in space instead of adding it to the square of the separation in time to find the interval because I am picturing interval as the hypotenuse of a right triangle with legs that are separation in time and separation in space.

And another thing... I'm having trouble understanding why the closer you get to the speed of light (let's say on a journey to another galaxy), the slower time actually goes for you, but not for others who stayed on Earth.

Aaand one more... How can events that are separated in space and time not be separated in spacetime because interval is zero? If they happen 6 meters apart and 6 seconds apart, then interval would be zero, right? But they ARE separated in space and they ARE separated in time, so it just doesn't make sense.

Thanks! And please bear with me; I have not been learning this for more than a week or two :)

 

[ghwellsjr]

I think the easiest way to understand the spacetime interval is realize that depending on the two events in question, it is either a space interval (or distance) or a time interval (or period) or neither in the case where it is zero. In the case where it is a time interval, it can be measured with an inertial clock that is present at both events. In the case of a space interval, it can be measured with an inertial ruler where the two events are measured at the same time (according to Einstein's synchronization criterion). When the spacetime interval evaluates to zero, it cannot be measured with either a clock or a ruler and it is called a null interval.

The reason why you subtract the squares is so that if the time component is greater than the spatial components, you get a time interval or the other way around you get a space interval. If you added them, there would be no distinction between a space interval and a time interval and there would never be a null case.

 

[phinds]

And another thing... I'm having trouble understanding why the closer you get to the speed of light (let's say on a journey to another galaxy), the slower time actually goes for you, but not for others who stayed on Earth.

Time passes for you at exactly the same rate as it does for someone on Earth. One second per second. According to a distant observer, it passes more slowly for you but that's only fair since for you it passes more slowly for them. It is only if you get back together that you face the possibility that the two of you, having taken different paths through space-time, will no longer have aged by the same amount.

Aaand one more... How can events that are separated in space and time not be separated in spacetime because interval is zero? If they happen 6 meters apart and 6 seconds apart, then interval would be zero, right? But they ARE separated in space and they ARE separated in time, so it just doesn't make sense.

I'm not sure what you are asking. Events that are separated by 6 meters and 6 seconds ARE separated in space-time. Are you sure it was stated that they would not be?

Thanks! And please bear with me; I have not been learning this for more than a week or two :)

Sounds like you're progressing nicely. Keep it up.

 

[ghwellsjr]

Also, you should be aware that the closer you get to the speed of light, time is going slower for you in the rest frame of the Earth but time is going slower for Earth in your rest frame. Neither of you can detect that time is anything other than perfectly normal.

 

[ghwellsjr]

One other thing, your example of 6 meters and 6 seconds is not what you wanted. It should be 6 seconds and 6 light-seconds. That is a null interval which cannot be measured with either a clock or a ruler.

 

[vela]

What I don't understand is why you subtract the squares of the distance in space instead of adding it to the square of the separation in time to find the interval because I am picturing interval as the hypotenuse of a right triangle with legs that are separation in time and separation in space.

It's a consequence of the geometry of spacetime. If separations were measured as $s^2=t^2+x^2$, spacetime would be Euclidean. In our universe, however, we find that $s^2=x^2-t^2$ accurately describes the geometry of spacetime. Spacetime is what's called a Minkowski space.

 

[mastercoin]

Are there any imtervals when I capture light?

I recall watching a lecture on TED talks by Thad Cochran stating space and time could be superimposed and I thought this would eliminate any possible intervals

 

[phinds]

Are there any imtervals when I capture light?

I'm not even sure what that question MEANS. If you capture some light and then wait a second and capture some more light, then those are two different events.

I recall watching a lecture on TED talks by Thad Cochran stating space and time could be superimposed and I thought this would eliminate any possible intervals

Again, I don't understand what you are talking about. Space-time is a framework in which things happen. Time is what keeps it from all happening at once and space is what keeps it from all happening right in my chair here where I'm minding my own business.

 

[ghwellsjr]

Are there any imtervals when I capture light?

I recall watching a lecture on TED talks by Thad Cochran stating space and time could be superimposed and I thought this would eliminate any possible intervals

The spacetime interval for light is the null case. It cannot be measured with either a clock or a ruler. Its spacetime interval is zero but not zero distance or zero time. It is neither.

I don't know what Cochran was talking about. It would be helpful if you would provide a link.

 

[cuchey615]

One other thing, your example of 6 meters and 6 seconds is not what you wanted. It should be 6 seconds and 6 light-seconds. That is a null interval which cannot be measured with either a clock or a ruler.

Oh ok that makes sense. So the term "null interval" doesn't actually mean there is no interval, it would just mean that we can't measure it?

 

[robphy]

Oh ok that makes sense. So the term "null interval" doesn't actually mean there is no interval, it would just mean that we can't measure it?

It means that the interval is assigned a value of zero, even though the interval itself is a segment (not a point) in spacetime.
A wristwatch-wearing observer who can send and receive light signals can measure it.

 

[ghwellsjr]

It means that the interval is assigned a value of zero, even though the interval itself is a segment (not a point) in spacetime.
A wristwatch-wearing observer who can send and receive light signals can measure it.

Please explain how.

 

[robphy]

The wristwatch-wearing observer can use the light-signals to do radar measurements (in a small region of spacetime).
Let P be an event on that inertial observer's worldline, and Q is a distant event.
To measure the square-interval, arrange to send a light-signal to Q from an appropriate event S[end] on his worldline and wait for its radar echo at event R[eceive] on his worldline. (S and R are intersections of the worldline with the light-cone of Q.)
Define the square-interval as $\Delta s^2_{PQ}=(t_R-t_P)(t_S-t_P),$
a formula due to A.A.Robb (1911) [also found in Synge and in Geroch],
where $t_S$ is the wristwatch-time of event S, etc...
If $\Delta s^2_{PQ}>0$, the PQ is timelike.
If $\Delta s^2_{PQ}<0$, the PQ is spacelike.
If $\Delta s^2_{PQ}=0$, the PQ is lightlike (null).
If PQ is null,
then
if it is future-directed, then P coincides with S,
else if it is past-directed, then P coincides with R,
otherwise Q is on the worldline and P,Q,R,S all coincide.

By defining $\Delta t_{PQ}=(t_R+t_S)/2$ and $\Delta x_{PQ}=(t_R-t_S)/2$,
you get back the more familiar formula $\Delta s^2_{PQ}=\Delta t^2_{PQ}-\Delta x^2_{PQ}$.

Note that this wristwatch is not present at Q...

In addition, this wristwatch cannot be taken on a trip (with another observer) from P to Q if PQ is not timelike.

With a little more effort, one can measure the square interval of PQ where both P and Q are distant events not the observer's worldline.

 

[DrGreg]

Define the square-interval as Δs2PQ=(tR−tP)(tS−tP),\Delta s^2_{PQ}=(t_R-t_P)(t_S-t_P),
a formula due to A.A.Robb (1911) [also found in Synge and in Geroch],

@ghwellsjr: You may be interested in more details about this method (for the more general case when both P and Q are distant events) in an old post of mine: Lorentz interval #3.