# What do we mean by saying that one clock had been "running slower" than another clock; or that two clocks had been "running equally"?

Several posts on this site, especially on the topic of relativity, refer to comparisons between clocks in terms of their "running";
one having "run faster" than the other, one having "run slower" than the other,
or both having "run equally (fast)".

What exactly do we mean by saying that

• clock $A$ had "run slower" than clock $B$, or that

• clocks $A$ and $B$ had "run equally",

in applicable trials ?

Given clock $A := (\mathcal A, t_A)$, where $\mathcal A$ denotes the ordered set of clock $A$'s indications ("positions of the little hand" etc.) and $t_A : \mathcal A \rightarrow \mathbb R$ denotes the corresponding readings of clock $A$,
and correspondingly given clock $B := (\mathcal B, t_B)$,

and considering two specific trials (separately one of clock $A$, and one of clock $B$) where

• $A_{\circ J}$ is the indication of clock $A$ at the beginning of its trial,

• $A_{\circ K}$ is the indication of clock $A$ at the end of its trial,

• $B_{\circ P}$ is the indication of clock $B$ at the beginning of its trial,

• $B_{\circ Q}$ is the indication of clock $B$ at the end of its trial,

then

clock $A$ is said to have been running slower than clock $B$ (and correspondingly, clock $B$ is said to have been running faster than clock $A$) in these trials if

$$\frac{t_A[~A_{\circ K}~] - t_A[~A_{\circ J}~]}{\Delta \tau A[~{}_{\circ J}, {}_{\circ K}~]} \lt \frac{t_B[~B_{\circ Q}~] - t_B[~B_{\circ P}~]}{\Delta \tau B[~{}_{\circ P}, {}_{\circ Q}~]};$$

and clock $A$ and clock $B$ are said to have been running equally in these trials if

$$\frac{t_A[~A_{\circ K}~] - t_A[~A_{\circ J}~]}{\Delta \tau A[~{}_{\circ J}, {}_{\circ K}~]} = \frac{t_B[~B_{\circ Q}~] - t_B[~B_{\circ P}~]}{\Delta \tau B[~{}_{\circ P}, {}_{\circ Q}~]};$$

where $\Delta \tau A[~{}_{\circ J}, {}_{\circ K}~]$ denotes clock $A$'s duration from having indicated the beginning of its trial until having indicated the end of its trial,
and $\Delta \tau B[~{}_{\circ P}, {}_{\circ Q}~]$ denotes clock $B$'s duration from having indicated the beginning of its trial until having indicated the end of its trial.

• Thumbs up for showing just how hard mathematicians manage to work to make a trivial statement hard to read. Sep 9 '15 at 18:20
• @CuriousOne: "Thumbs up for showing just how hard mathematicians manage to work to make a trivial statement hard to read." -- Seems to take a mathematician (like yourself?) to call "trivial" the need to compare durations, $\Delta \tau A[~...~]$ and $\Delta \tau B[~...~]$, of two participants. Sep 9 '15 at 18:30
• Don't worry, you are not even close to getting my "top mathematical obfuscator" price. That goes to a person who wrote a 400 page dissertation about communication protocols in computer science. Except for the foreword, the citations and index pages and the occasional "Lemma:" and "Proof:" it was entirely written in hieroglyphics that someone had invented as a shortcut notation. That thing was fun... and pretty much all of us thought that the person only got their PhD because the professor who had to grade it didn't want to admit that they could not read a single sentence. :-) Sep 9 '15 at 20:42
• @CuriousOne: "Don't worry [...]" -- What, me worry?? ... How many times have I quoted "All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points." yet? ... Sep 9 '15 at 21:06