special relativity - Is there always a frame in which spatially separated events are simultaneous?

In relativity, if two events are simultaneous in a specified frame, they cannot be simultaneous in any other frame.

My question is this: given any two events, is there always a frame in which these two events are simultaneous? For example, if I drop a blue ball on one side of a tennis court, and my friend drops a red ball on the opposite side of the court one day later -- from my frame one day later -- is there a frame in which the blue and red balls hit the ground simultaneously?

Answer

Is there always a frame in which spatially separated events are simultaneous?

The answer is no.

Two events that are spatially separated in one frame of reference

(1) will be co-located in another frame of reference and not simultaneous in any frame if the interval is time-like

(2) will be simultaneous in another frame of reference and not co-located in any frame if the interval is space-like .

(3) will be neither co-located nor simultaneous in any other frame if the interval is light-like.

Time-like interval

If the interval is time-like, the separation in time, $|c\Delta t|$, is larger than the separation in space $|\Delta x|$:

$$|c\Delta t| \gt |\Delta x|$$

Thus, there is a frame of reference in which $\Delta x' = 0$; the two events are co-located in this frame.

Space-like interval

If the interval is space-like, the separation in time is less than the separation in space:

$$|c\Delta t| \lt |\Delta x|$$

Thus, there is a frame of reference in which $c\Delta t' = 0$; the two events are simultaneous in this frame.

Light-like interval

If the interval is light-like the separation in time equals the separation in space:

$$|c\Delta t| = |\Delta x|$$

Thus, in all frames of reference, the events are neither co-located nor simultaneous, i.e.,

$$|c\Delta t'| = |\Delta x'|$$

All of this follows directly from the Lorentz transformation. Let's take your example of two events with spatial separation of a tennis court so

$$|\Delta x| = 78\mathrm m$$

Light travels this distance in $\Delta t_c = \frac{78}{300 \cdot 10^6} = 260\mathrm{ns}$

Thus, if the two events occur within 260ns in this frame of reference, the events have space-like interval and are thus simultaneous in another, relatively moving reference frame of reference.

Since, in your example, the events occur 1 day apart, the events have time-like interval and cannot be simultaneous in any reference frame.