Definition of relativity's interval

[Severian596]

Definition of relativity's "interval"

Okay I have Kip Thorne's text in front of me that says the following:

If $$(\Delta s)^2 > 0$$ the events are called spacelike
If $$(\Delta s)^2 = 0$$ the events are called lightlike
If $$(\Delta s)^2 < 0$$ the events are called timelike

Then I have David W. Hogg's text in front of me that says:

If $$(\Delta s)^2 > 0$$ the events are called timelike
If $$(\Delta s)^2 = 0$$ the events are called lightlike
If $$(\Delta s)^2 < 0$$ the events are called spacelike

No joke. Which one's right? Understanding of this cursed interval was just dawning on me when I saw this. I was coming to the conclusion that $$\Delta s$$ is the distance light traveled in the frame of reference minus the distance between the events in that frame. I was thinking if this value is positive, light traveled farther than the distance between the events, thus there is a valid space-like interval.

If you don't mind could you also clear up what the sam heck this interval means?! I'm not sure I'm getting it.

TIA

 

[GRQC]

Originally posted by Severian596
Okay I have Kip Thorne's text in front of me that says the following:

If $$(\Delta s)^2 > 0$$ the events are called spacelike
If $$(\Delta s)^2 = 0$$ the events are called lightlike
If $$(\Delta s)^2 < 0$$ the events are called timelike

Then I have David W. Hogg's text in front of me that says:

If $$(\Delta s)^2 > 0$$ the events are called timelike
If $$(\Delta s)^2 = 0$$ the events are called lightlike
If $$(\Delta s)^2 < 0$$ the events are called spacelike

No joke. Which one's right?

It depends on how you define the signature of the metric. For example, the Minkowski metric can be written

$$ds^2 = -dt^2 + dx^2+dy^2+dz^2$$

or equivalently it can be

$$ds^2 = +dt^2 - dx^2-dy^2-dz^2$$

It's all really a matter of preference.

 

[Severian596]

Originally posted by GRQC
It depends on how you define the signature of the metric.

Thank you very much, that pinpoints the inconsistencey. Thorne defines the metric as $$ds^2 = -dt^2 + dx^2+dy^2+dz^2$$, where Hogg defines it with an opposite sign. And thank you for the use of the term "Minkowski metric." I was not aware that it is the technical term for interval.

What's the significance of the Minkowski Metric? If for example we define two events as having spacelike intervals, what does that mean? I want to say that it means they could influence each other since they'd exist within each other's light cones, where a timelike interval means the events are outside each others' light cones.

 

[EL]

Originally posted by Severian596
What's the significance of the Minkowski Metric? If for example we define two events as having spacelike intervals, what does that mean? I want to say that it means they could influence each other since they'd exist within each other's light cones, where a timelike interval means the events are outside each others' light cones.

It's the other way...

 

[Severian596]

So spacelike means two events are outside each other's light cones?

 

[pmb_phy]

Originally posted by Severian596

What's the significance of the Minkowski Metric? If for example we define two events as having spacelike intervals, what does that mean? I want to say that it means they could influence each other since they'd exist within each other's light cones, where a timelike interval means the events are outside each others' light cones.

The Minkowski metric pertains to what is called Lorentz coordinates, i.e. the system of coordinates whose spatial coordinates are Cartesian (x, y, z) and which pertains to an inertial frame of reference.

As far as timelike/spacelike/lightlike - To see this cleary think of a (ct,x) spacetime diagram corresponding to an inertial frame of reference. You'll notice that the worldline corresponding to a luxon (particles for which v = c) has a slope of 1. The worldine of a tardyons (particles for which v < c) has a slope greater than 1 and the worldline corresponding to a tachyon (particles for which v > c) has a slope of less than one.

If you have two events with a timelike spacetime separation then a tardyon can be present at both events.

If you have two events with a lightlike spacetime separation then a luxon can be present at both events.

If you have two events with a spacelike spacetime separation then a tachyon can be present at both events.

 

[Severian596]

Strange way to put it, but I got it! Thanks very much.

 

[pmb_phy]

Originally posted by Severian596
Strange way to put it, but I got it! Thanks very much.

The reason I phrased it as such was that it doesn't depend on any convension for the Minkowski metric if you've noticed.

 

[EL]

Originally posted by pmb_phy
The worldine of a tardyons (particles for which v = c) has a slope greater than 1

Should be v<c...

 

[pmb_phy]

Originally posted by EL
Should be v<c...

Whoops! Thanks EL!

 

[jcsd]

The important thing about spacetime intervals is that they are Lorentz invariant and therefore the same for all observers. If you mutiply the Minowski metric by a coefficient (in the above examples -1) it will not affect it's Lorentz invariance, so it becomes a matter of convention and taste as to which form you use.