Example of space-like intervals in spacetime

Question

From wikipedia:

When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time, but there is no reference frame in which the two events can occur in the same spatial location.

Can someone give me an example of this?

Answer 1

A very simple example is the following:

Two observers A and B are spatially separated (one in London, the other in Cambridge). They are static with respect to each other and can therefore measure the same time. So, we can fix a particular time T.

The two events $(T,\mathbf{x_{London}})$ and $(T,\mathbf{x_{Cambridge}})$ are spatially separated - one cannot influence the other. Anything happening at $(T,\mathbf{x_{London}})$, for example, can only influence B later on when it crosses its light cone at $T+\frac{||\mathbf{x_{Cambridge}}-\mathbf{x_{London}}||}{c}$.

Answer 2

Ultimately the question is about the causal structure of geometry. The local, causal structure of any event $E$ (represented by a point in Minkowski space) is determined by its light-cone, the boundary between events - past and future - that are spacelike or timelike w.r.t $E$.

In this figure the vertical axis is time (t) and the two horizontal axes are spatial (x,y). The points inside the cone are those which are in the causal past and future of the event $E$. The points lying outside the cone cannot have any causal influence on or cannot be causally affected by $E$. If you pick any point $P =(x,y)$ in this region, then the interval between $E = (0,0)$ and $P$ will be "spacelike". The interval between any two points in Minkowski space is calculated using the (-,+,+) metric:

$$ l_{12}^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 - c^2(t_1 - t_2)^2 $$

where $c$ is the speed of light. Note the minus sign. That is what distinguishes timelike dimensions from spacelike ones. Points lying on the light-cone itself have zero or "null" separation in this metric.

To answer your question, pick any two points $(x_1,y_1,t_1)$ and $(x_2,y_2,t_2)$ such that $l_{12}^2 \gt 0$. These two points have a space-like separation. As for a concrete example of such a pair of events, consider any two clocks in your house with spatial separation $\delta x$ between them, synchronized to within $\delta t$ seconds of each other such that $\delta x \gt c \delta t$. Then everytime the clocks tick, the separation between those two events in spacelike.

Note: The sign associated with spacelike or timelike intervals is a matter of convention and depends on your choice of metric $(-,+,+)$ or (+,-,-)

Answer 3

Recipe to generate events with space like separation (and be able to prove it):

1. Set up a radio transmitter at point $A$.
2. At point $B$ and $C$ which are arrange so that A lies at the center point of the segment between them we put radio receivers that will perform some action on receiving a radio pulse from A. We make the distance between them greater than the speed-of-light * however-long-it-takes-the gadget to act.
3. Emit a pulse from the transmitter at $A$

The result is that two events will happen---one each at $B$ and $C$---which appear to be simultaneous as observed from A. Thus they have space-like separation.