0. Summary and Outline of the following derivations
"Can [...] space-like interval invariance be demonstrated without using two spatial dimensions?"
Yes1.
$\def\ReceivedAndReflected{{\hphantom{.}\text r \hphantom{.}}\llap{\bigcirc}}$
$\def\SimultaneousTo{{\hphantom{.}\text s \hphantom{.}}\llap{\bigcirc}}$
$\def\CoincidentWith{{\hphantom{.}\text c \hphantom{.}}\llap{\bigcirc}}$
The proof presented below (completed in section 3) uses two lemmas:
- the comparison of a distance between (any) two members ($A$ and $B$) of one inertial system and the distance between two (suitably selected) members ($P$ and $Q$) of another (suitable) inertial system, selected such that these two pairs are "moving uniformly along each other" (in order to fit into a 1+1-dimensional description), and with $P$ and $Q$ additionally selected such that $A$'s indication of having met and passed $P$ was simultaneous to $B$'s indication of having met and passed $Q$; symbolically: $A_P \SimultaneousTo B_Q$.
As to be expected (for a derivation of what's a.k.a. "length contraction"), the ratio between these distances is obtained as
$$\left( \frac{AB}{PQ} \right) = \sqrt{1 - \beta^2},$$
where $c \, \beta$ denotes the constant and mutually equal speed of all members of the one inertial system wrt. members of the other. Further:
- the comparison of certain durations of members of these two inertial systems who are in suitable relation to each other; in particular deriving the ratio
$$ \left( \frac{\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith J \, ]}}{\tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo J \CoincidentWith A \, ]}} \right) $$
between $A$'s duration from its indications of having met and passed $P$ until its indication of having met and passed $J$ and $P$'s duration from its indication of having met and passed $A$ until its indications simultaneous to $J$'s of having met and passed $A$.
As to be expected (for a derivation of what's a.k.a. "time dilation, in SR"), the ratio between these durations is obtained as
$$ \left( \frac{\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith J \, ]}}{\tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo J \CoincidentWith A \, ]}} \right) = \sqrt{1 - \beta^2}. $$
These derivations are accompanied by some suggestive illustrations ("one-dimensional snapshots") as memory aides, but presently without much additional verbiage (in order for the length of my answer to be not too excessive).
In section 3 follows a proof of invariance of spacelike interval $s^2[ \, \varepsilon_{AP}, \varepsilon_{BQ} \, ]$ as representative of general cases. Section 4 concludes with a corresponding proof of invariance of timelike interval $s^2[ \, \varepsilon_{AP}, \varepsilon_{AK} \, ]$.
The derivations and proofs are coordinate-free, and notation has been chosen accordingly. The present derivation steps may not yet be optimal ...
1: Well -- Yes, provided that inertial systems can be considered given; i.e. by W. Rindler's characterization:
"An inertial frame is simply an infinite set of point particles sitting still in space relative to each other."; such that a (non-zero) distance can be attributed to any pair of such point particles who are members of the same inertial system.
However: a (comprehensible, satisfying) demonstration of exactly which point particles are "sitting still relative to each other", if at all, is likely not available by considering only 1+1 dimensions, i.e. only one spatial dimension along with one temporal dimension.
1. Comparison of distances between inertial systems
$$\matrix{ :::::::::::::::::: \cr \, \cr ::::: K ::::::: } \! \! \matrix{ ::::::: A === \cr ~~~~ \varepsilon {\small \longrightarrow} \cr ::::::: P :::::::: } \! \! \matrix{ ===== M ====== \cr \, \cr :::::::::::::::::::::::::::::::::::: } \! \! \matrix{ == B ::: \cr ~~~\varepsilon \cr :::::: Q ::: } \! \! \matrix{ ::::::: F :::::: \cr \, \cr :::: \hphantom{ :: F :::::: } }$$
$$ \left( \frac{AB}{AF} \right) + \left( \frac{BF}{AF} \right) = 1, $$
$$ \left( \frac{KP}{KQ} \right) + \left( \frac{PQ}{KQ} \right) = 1 \tag{1a} $$
$$\matrix{ ::::::: A :::::::: \cr \varepsilon ~~ \cr :::::: K === } \! \! \matrix{ ::::::::::::::: \cr \, \cr == P == } \! \! \matrix{ :::::::::::::::::::::::::::::: B :: \cr \, \cr === N ======== } \! \! \matrix{ :::::::: F ::::: \cr {\small \longrightarrow} \varepsilon ~ \cr === Q :::: } \! \! \matrix{ ::: \cr \, \, \cr ::: }$$
$$\beta := \beta_{ABF}[ \, Q \, ] = \left( \frac{BF}{AF} \right), $$
$$ \beta := \beta_{KPQ}[ \, A \, ] = \left( \frac{KP}{PQ} \right) = \left( \frac{KP}{KQ} \right) / \left( \frac{PQ}{KQ} \right). \tag{1b} $$
$$ \left( \frac{AB}{AF} \right) = (1 – \beta), $$
$$ \left( \frac{PQ}{KQ} \right) = 1 – \left( \frac{KP}{KQ} \right) = 1 – \left( \frac{KP}{PQ} \right) \, \left( \frac{PQ}{KQ} \right) = 1 – \beta \, \left( \frac{PQ}{KQ} \right) = \frac{1}{(1 + \beta)}. \tag{1c} $$
The decisive requirement of reciprocity of corresponding distance ratios, ensuring mutual comprehensibility and agreeability of the comparison:
$$ \left( \frac{AB}{PQ} \right) = \left( \frac{KQ}{AF} \right). \tag{1d} $$
$$ \left( \frac{AB}{PQ} \right) = \sqrt{ \left( \frac{AB}{PQ} \right) \, \left( \frac{KQ}{AF} \right) } = \sqrt{ \left( \frac{AB}{AF} \right) \, \left( \frac{KQ}{PQ} \right) } = \sqrt{ (1 – \beta) \, (1 + \beta) } = \sqrt{ 1 - \beta^2 }.
%\tag{1e}
$$
2. Comparison of durations between inertial systems
$$\matrix{ \hphantom{ ::::: J :::::::::: } \cr \, \cr ::::: J :::::::::: } \! \! \matrix{ ::::: A ::::::: \cr \hphantom{ \rightarrow } \varepsilon {\small \longrightarrow} \cr ::::: P ::::::: } \! \! \matrix{ ::::: B ::::::: \cr \, \cr :::::::::::::::: } \! \! \matrix{ :::::::::::::::::::: G ::::: \cr \, \cr ::::::::: U ::::::\hphantom{ ::::::::: } }$$
$$ \left( \frac{AB}{AG} \right) + \left( \frac{BG}{AG} \right) = 1, $$
$$\left( \frac{JP}{JU} \right) + \left( \frac{PU}{JU} \right) = 1. \tag{2a}$$
$$\matrix{ \hphantom{ :::::::: } ::::: A :::::::::::: B :::::::::::::::::::: \cr \, \cr ::::: J ::::::::::::::: P ::::::::::::::::::::::::::: } \! \! \matrix{ ::::::: G :::::::: \cr \leftarrow \! \! \! \! \leftarrow \! \! \! \rightarrow \varepsilon \hphantom{ \longrightarrow ~ } \cr ::::::: U ::::::: }$$
Subsequently:
$$\matrix{ \hphantom{:::} ::::: A ::::: \cr \, \cr ::::: J ::::::::: } \! \! \matrix{ ::::::: B :::::::: \cr \leftarrow \! \! \! \! \leftarrow \! \! \! \! \leftarrow \varepsilon \leftarrow \! \! \! \! \leftarrow ~\cr ::::::: P :::::::: } \! \! \matrix{ :::::::::::::::::: G :::::: \hphantom{ ::::::::: } \cr \, \cr ::::::::::::::::::::::::::: U ::::::: }$$
And finally:
$$\matrix{ ::::: A ::::::::: \cr \phantom{ ~~~~ } \varepsilon \leftarrow \! \! \! \! \leftarrow \! \! \! \! \leftarrow \cr ::::: J ::::::::: } \! \! \matrix{ ::: B :::::::::::: \cr \, \cr :::::::: P ::::::: } \! \! \matrix{ :::::::::::::: G :::::::::: \hphantom{ ::::::::: } \cr \, \cr ::::::::::::::::::::::::::: U ::::::: }$$
$$\beta := \beta_{ABG}[ \, P \, ] := \frac{1}{ \left( \frac{AG}{AB} \right) + \left( \frac{BG}{AB} \right) } = \frac{\left( \frac{AB}{AG} \right)}{1 + \left( \frac{AB}{AG} \right) \left( \frac{BG}{AB} \right) } = \frac{1 – \left( \frac{BG}{AG} \right)}{1 + \left( \frac{BG}{AG} \right)},$$
$$\beta := \beta_{JPU}[ \, A \, ] := \frac{1}{ \left( \frac{PU}{JP} \right) + \left( \frac{JU}{JP} \right) } = \frac{\left( \frac{JP}{JU} \right)}{\left( \frac{JP}{JU} \right) \left( \frac{PU}{JP} \right) + 1 } = \frac{1 – \left( \frac{PU}{JU} \right)}{1 + \left( \frac{PU}{JU} \right)} = \frac{\left( \frac{JU}{PU} \right) – 1}{\left( \frac{JU}{PU} \right) + 1}.$$
The decisive requirement of reciprocity of corresponding duration ratios, ensuring mutual comprehensibility and agreeability of the comparison:
$$\large \left( \frac{\tau^P_{[ \, \CoincidentWith A, \, \CoincidentWith B \, ]}}{\tau^A_{[ \, \CoincidentWith P, \, \SimultaneousTo B \CoincidentWith P \, ]}} \right) = \left( \frac{\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith J \, ]}}{\tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo J \CoincidentWith A \, ]}} \right). \tag{2b} $$
$$\left( \frac{\tau^P_{[ \, \CoincidentWith A, \, \CoincidentWith B \, ]}}{\tau^A_{[ \, \CoincidentWith P, \, \SimultaneousTo B \CoincidentWith P \, ]}} \right) := \frac{2 \, PU / c}{(AG + BG) / c} := \frac{2}{ \left( \frac{AG}{PU} \right) + \left( \frac{BG}{PU} \right) },$$
$$\left( \frac{\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith J \, ]}}{\tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo J \CoincidentWith A \, ]}} \right) := \frac{2 \, AG / c}{(PU + JU) / c} := \frac{2}{ \left( \frac{PU}{AG} \right) + \left( \frac{JU}{AG} \right) }.$$
$$\eqalign{ \frac{2}{ \left( \frac{AG}{PU} \right) + \left( \frac{BG}{PU} \right) } &= \frac{2}{ \left( \frac{PU}{AG} \right) + \left( \frac{JU}{AG} \right) } = \left( \frac{AG}{PU} \right) \, \left( \frac{2}{ 1 + \left( \frac{JU}{PU} \right) } \right) \\ &= \left( \frac{AG}{PU} \right) \, \left( \frac{ 1 + \left( \frac{BG}{AG} \right) }{2} \right) \, \left( \frac{2}{ 1 + \left( \frac{BG}{AG} \right) } \right) \, \left( \frac{2}{ 1 + \left( \frac{JU}{PU} \right) } \right) \\ &= \left( \frac{ \left( \frac{AG}{PU} \right) + \left( \frac{BG}{PU} \right) }{2} \right) \, \left( 1 + \beta \right) \, \left( 1 – \beta \right) \\ &= \left( \frac{ \left( \frac{AG}{PU} \right) + \left( \frac{BG}{PU} \right) }{2} \right) \, \left( 1 – \beta^2 \right) \\ &= \sqrt{ 1 – \beta^2 }. }$$
$$\large \left( \frac{\tau^P_{[ \, \CoincidentWith A, \, \CoincidentWith B \, ]}}{\tau^A_{[ \, \CoincidentWith P, \, \SimultaneousTo B \CoincidentWith P \, ]}} \right) = \left( \frac{\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith J \, ]}}{\tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo J \CoincidentWith A \, ]}} \right) = \sqrt{ 1 – \beta^2 }. \tag{2c}$$
3. Invariance of spacelike intervals
$$ s^2[ \, \varepsilon_{AP}, \varepsilon_{BQ} \, ] := AB^2 \, \overset{?}{=} \, PQ^2 - \left( c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo Q \CoincidentWith B \, ]} \right)^2. \tag{3a} $$
$$ \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo Q \CoincidentWith B \, ]} = \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo Q \CoincidentWith F \, ]} - \tau^Q_{[ \, \CoincidentWith B, \, \CoincidentWith F \, ]} = \left( \frac{1}{2} \right) \, \tau^P_{[ \, \CoincidentWith A, \, \ReceivedAndReflected Q \ReceivedAndReflected P \CoincidentWith A \, ]} - \sqrt{ 1 - \beta^2 } \, \tau^B_{[ \, \CoincidentWith Q, \, \SimultaneousTo F \CoincidentWith Q \, ]},$$
$$ c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo Q \CoincidentWith B \, ]} = \left( \frac{c}{2} \right) \, \tau^P_{[ \, \CoincidentWith A, \, \ReceivedAndReflected Q \ReceivedAndReflected P \CoincidentWith A \, ]} - c \, \sqrt{ 1 - \beta^2 } \, \tau^B_{[ \, \CoincidentWith Q, \, \SimultaneousTo F \CoincidentWith Q \, ]} = PQ - \sqrt{ 1 - \beta^2 } \, \frac{BF}{\beta}.
%\tag{3b}
$$
$$ AB^2 \, \overset{?}{=} \, PQ^2 - \left( PQ - \sqrt{ 1 - \beta^2 } \, \frac{BF}{\beta} \right)^2 = \frac{BF^2}{\beta^2} - BF^2 - 2 \, PQ \, \sqrt{ 1 - \beta^2 } \, \frac{BF}{\beta}. \tag{3c} $$
$$PQ = \frac{AB}{\sqrt{1 - \beta^2}},$$
$$ AB^2 \, \overset{?}{=} \, \frac{BF^2}{\beta^2} - BF^2 - \frac{2 \, AB \, BF}{\beta}. \tag{3d}$$
$$\frac{1}{\beta} \overset{?}{=} \left( \frac{AB}{BF} \right) + 1 = \left( \frac{AB + BF}{BF} \right) = \left( \frac{AF}{BF} \right) \tag{3e} $$
which holds according to eq. (1a). Therefore eq. (3a) holds as well: the invariance of spacelike interval $s^2[ \, \varepsilon_{AP}, \varepsilon_{BQ} \, ]$, and consequently invariance of spacelike intervals in general.
4. Invariance of timelike intervals
$$ s^2[ \, \varepsilon_{AP}, \varepsilon_{AK} \, ] := -\left( c \, \tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith K \, ]} \right)^2 \, \overset{?}{=} \, KP^2 - \left( c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo K \CoincidentWith A \, ]} \right)^2. \tag{4a} $$
$$ \tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith K \, ]} = \sqrt{ 1 - \beta^2 } \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo K \CoincidentWith A \, ]}, $$
$$ \beta = \frac{KP}{c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo K \CoincidentWith A \, ]}} := \beta_{KPQ}[ \, A \, ] = \frac{KP}{PQ} = \frac{KP}{c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo Q \CoincidentWith F \, ]}}. $$
Accordingly substituting $\tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith K \, ]}$ and $KP$, eq. (4a) is seen to be manifestly satisfied:
$$ -\left( c \, \tau^A_{[ \, \CoincidentWith P, \, \CoincidentWith K \, ]} \right)^2 = (\beta^2 - 1) \, \left( c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo K \CoincidentWith A \, ]} \right)^2 = KP^2 - \left( c \, \tau^P_{[ \, \CoincidentWith A, \, \SimultaneousTo K \CoincidentWith A \, ]} \right)^2. $$
Therefore the timelike interval $s^2[ \, \varepsilon_{AP}, \varepsilon_{AK} \, ]$ is invariant; and so are timelike intervals in general.
Best Answer
This problem would have given me fits when I first learned relativity, but since I've started using the geometric point of view as the tool I reach for first it is almost trivial.
The direction is easy: boost in the direction from the earlier to the later event as measured in your current frame. Why? Because you want the space-like axis to tilt upward in that direction.
Getting the speed is also surprisingly easy. You need the new space-like axis to have a slope (in your current coordinate system) equal to $(\Delta x)/(c \,\Delta t)$, which is exactly the $\beta$ of the boost you need.