# Space-like Events

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1. Jul 25, 2015

### Mikeal

I decided to re-visit special relativity and spacetime. All went well until I reached the subject of time-like and space-like events and their relation to causality.

Time-like Events

The residual spacetime value between two time-like events is defined as: ∆S = √(c∆t)^ - ∆x^2, where c∆t > ∆x.

Thus, ∆S is real and defines as series of hyperbolic curves in the past and future light-cones of an Event (0,0). Events in these regions can have a causal relationship with Event (0,0).

Proper time is given by ∆S/c = ∆t = t√(1 - (V^2/c^2))

Any attempt to determine proper distance in these regions, results in an imaginary value (i.e. there is no proper distance for space-like separated events). So far, so good.

Space-like Events

This is where my understanding breaks down.

The residual spacetime value between two space-like events is defined as: ∆S = √(c∆t)^2 - ∆x^2, where c∆t < ∆x.

Thus, ∆S is imaginary. This makes sense, as space-like events are in the "elsewhere regions" of Event (0,0) and thus cannot have a causal relationship with it.

However, The texts that I've read , magically reverse the signs within the square root, such that:

∆S = √∆x^2 - (c∆t)^2 and ∆S thus becomes real. I think this mathematical manipulation is akin to rotating the elsewhere regions by ninety degrees to where the past and future lights cones used to be.

Once this is done: ∆S = ∆x√(1 - (c2/V2)), which is defined as the "proper distance" between Event (0,0) and events in the elsewhere regions. Because of this mathematical manipulation, even though V > c, the proper distance is a real value. In fact if V < c, proper distance becomes imaginary.

So my questions are:

1) How can the spacetime relationship between ct and x be flipped at will to make things work-out right?

2) How can there be a "proper distance" between Event (0,0) and events in the elsewhere regions, if they are not causally related?

3) Shouldn't we just define the "elsewhere regions" as imaginary/not causally related and ignore the concept of "proper distance"?

2. Jul 25, 2015

### Mentz114

Here is an example (first posted by JDoolin). If an observer at (0,0) sends two light pulses, one each of the -x and x directions where the light is reflected back from (0,-5) and (0,5). The two mirrors are space-like separated but are still involved in an interaction with the emitter.

We can't ignore proper distance - it is the geometric invariant of the Minkowski spacetime.

3. Jul 25, 2015

### Staff: Mentor

So this is a piece of special relativity that is really clarified by the math of general relativity: Riemannian geometry.

In Riemannian geometry the signature of the metric determines the number of dimensions as well as whether they are spacelike or timelike. So the first expression you wrote corresponds to a (+---) signature, and the second one you wrote corresponds to a (-+++) signature.

The two signatures are completely equivalent. The thing which identifies time is the fact that there is only one timelike dimension, and whether you assign that a positive or negative signature doesn't matter.

4. Jul 26, 2015

### Mikeal

Events (0,-5) and (0,5) reside -5 and +5 space units to the left and right of Event (0,0), but zero time from it. In other words they are simultaneous with Event (0,0). My understanding is that they cannot have a causal relationship with Event (0,0), or each other, due to the finite speed of light.

5. Jul 26, 2015

### Mentz114

I was reacting to this
which I mis-read as a 'ignore the elsewhere regions'.

I'm not sure what 'causal relationship' is. if I can send a signal to something that is space-like removed from me (at transmission time), and be sure that it will arrive sometime in both our futures - are we causally connected ?

6. Jul 26, 2015

### DrGreg

This doesn't make sense. "Timelike" separation and "spacelike" separation applies to pairs of events in spacetime. An event has both a time and a place. Objects are not events. Places are not events. So you cannot talk about "spacelike separated places" or "spacelike separated objects".

You can't send a signal towards an event that is spacelike-separated from you. You can send a signal to an event in the future. You can send a signal to a place (without specifying any time).

7. Jul 26, 2015

### Mentz114

I was referring to pairs of events. I expressed it as
"if I can send a signal to something that is space-like removed from me (at transmission time)". So there is 'I' at (0,0) and the mirror at (0,5) . I can send a light pulse in the x-direction, which is a well-defined thing. How is that not towards the mirror ?

I may be sloppy in expressing things but not that sloppy.

Last edited: Jul 26, 2015
8. Jul 26, 2015

### Stephanus

Yes, I understand what you mean. Perhaps a little correction?
You send definetely toward the mirror. Not at (0,5) but at (5,5). Still (??,5) if I may say.

9. Jul 26, 2015

### Mentz114

No. I shine the light in the +x direction. There is no direction but +x or -x in the ST diagram.

Last edited: Jul 26, 2015