Generalizing Stewart's theorem to timelike triangles in Lorentzian spaces of constant curvature

Question

Stewart's theorem describes the length relations of the sides of a triangle and any one of its cevians in flat space ($n$-dimensional Euclidean space), and also in flat spacetime ($1 + n$-dimensional Minkowski space).

With names assigned as in the Wikipedia figure: $a, b, c$ (for the lengths) of triangle sides, $d$ of a cevian between side $a$ and the opposite vertex, and $n, m$ for the resulting segments of side $a$, Stewart's theorem is stated as

$$ b^2 \, m + c^2 \, n = a \, (d^2 + m \, n ),$$

of course together with

$$ a = n + m; $$

where in application to Minkowski space the squares of lengths are understood as suitably signed values of spacetime intervals $s^2$, and the "plain" lengths as correspondingly signed square roots $\text{Sgn}[ s^2 ] \, \sqrt{ s^2 \, \text{Sgn}[ s^2 ] }$.

I'm interested in corresponding relations of (suitably generalized) lengths, or in other words: in generalizations of Stewart's theorem, for triangles and their cevians in spaces and in (Lorentzian) spacetimes of constant curvature $k$; especially for their roles as model triangles in the characterization of $\text{CAT}[ \, k \, ]$ spaces by triangle comparison. (See also M. Kunziger, C. Sämann, "Lorentzian length spaces" (math:1711.08990), sect. 4.)

In spherical geometry the generalization is readily gotten; cmp. "Stewart theorem validity on a sphere" (mse/q/3696576). Assigning names of lengths again as above, together with $\theta$ and $\theta^{\prime}$, respectively, for the complementary angles along side $a$ of cevian $d$, opposite side $c$ and opposite side $b$, we have

$$ \begin{align*} \text{Cos}[ \, \theta \, ] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \, \right] &= \frac{\text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \, \right] \, - \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right]}{\text{Sin} \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \, \right]_{\phantom{y}}} & \\ &= \frac{\text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, - \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right]}{\text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right]} &= -\text{Cos}[ \, \theta^{\prime} \, ] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \, \right] \end{align*} $$

and therefore

$$ \begin{align} & \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \ \right] + \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] = \\ & \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \, \left( \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \ \right] + \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \ \right] \right) = \\ & \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right]. \end{align} $$

(Expanding the trigonometric functions in $\sqrt{ k } \ne 0$, terms up to second order cancel, the terms of third order sum up to being proportional to the above statement of Stewart's theorem, and remaining terms are of order five and higher.)

My question:
What are corresponding expressions, generalizing Stewart's theorem, for Lorentzian spaces of constant nonzero curvature ?

Note: Since generalized metric relations in Lorentzian spaces, a.k.a. spacetimes, $\mathcal S$, are often expressed in terms of Lorentzian distance$ \, \lambda : \mathcal S \times \mathcal S \rightarrow \{ \mathbb R_{(\ge 0)} \cup \infty \}$, which are strictly zero for all but timelike related pairs of events, the relations being sought may apply only to timelike triangles, i.e. with all sides being timelike, and their timelike or lightlike cevians.

Answer 1

My answer relies on certain formulas gathered from a proceedings article by Mariano Santander,
"The Hyperbolic-AntiDeSitter-DeSitter triality", Pub. de la RSME, Vol. 5 (2003), 247.
(Therefore my answer is not self-contained, unfortunately.)

As far as I understand, Santander's article identifies the relevant Lorentzian spaces of constant curvature as either "AntiDeSitter sphere", $\bf \text{AdS}^{1+1}$, or as "DeSitter sphere", $\bf \text{dS}^{1+1}$, in $1+1$ dimensions.

(In passing, Santander also provides corresponding "mathematical notation" for these spaces, namely $\bf \text{H}^2_1$ and $\bf \text{S}^2_1$, resp., which in turn appears to reasonably match the notation used for the model spaces of constant nonzero curvature in section 4.2 of the article by M. Kunziger, C. Sämann, "Lorentzian length spaces", linked in the OP question above.)

For these two spaces, Santander provides general cosine theorems in sect. 4 (however, without providing any explicit derivation I could recognize and reproduce). In terms of the notation of the OP question, where $a, b$ and $c$ are the names of the sides of a triangle and also stand for the corresponding values of Lorentzian distances, such that generally

$$ a \gt b + c, $$

$m$ and $n$ stand for the names of two segments of side $a$, as well as for the corresponding values of Lorentzian distances, whereby

$$ a = m + n, $$

and $d$ stands for the name as well as the value of Lorentzian distance of the cevian onto side $a$, whose footpoint separates $a$ into segments $n$ (complementing $b$) and $m$ (complementing $c$), such that (as one of two a special cases) generally

$$ b \gt n + d $$

(along with $ m \gt d + c$), these general cosine theorems can be specificly interpreted for instance

• for the "AntiDeSitter sphere":

$$ \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right] = \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] - \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Cosh}[ \, \angle^b_a \, ], \tag{1} \label{eq:1} $$

and

$$ \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] = \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] - \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] \, \text{Cosh}[ \, \angle^b_n \, ], \tag{2} \label{eq:2} $$

• or for the "DeSitter sphere":

$$ { \small \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right] = \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] \, \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] + \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] \, \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Cosh}[ \, \angle^b_a \, ],} \tag{3} \label{eq:3} $$

and

$$ { \small \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] = \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] + \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right] \, \text{Cosh}[ \, \angle^b_n \, ]. } \tag{4} \label{eq:4} $$

In any case, by construction, $\angle^b_n \equiv \angle^b_a$. Eliminating $\text{Cosh}[ \, \angle^b_n \, ] = \text{Cosh}[ \, \angle^b_a \, ]$ from equations $(\ref{eq:1})$ and $(\ref{eq:2})$ we have

$$ { \small \left( \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \, \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \, \right] - \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \, \right] \right) \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right] = \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } \\ { \small \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \, \right] \, \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right] - \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \right) \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \, \right].} $$

Inserting $a = m + n$ this reduces to

$$ { \small \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \, \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] = \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \, \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \, \right] + \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \, \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right]. } \tag{5} \label{eq:5} $$

Likewise Eliminating $\text{Cosh}[ \, \angle^b_n \, ] = \text{Cosh}[ \, \angle^b_a \, ]$ from equations $(\ref{eq:3})$ and $(\ref{eq:4})$ gives

$$\small \left( \! \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \, \right] \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] - \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \, \right] \right) \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right] = \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ \small \qquad \qquad \qquad \qquad \qquad \qquad \left( \! \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \, \right] \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \, \right] - \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \, \right] \right) \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \, \right]. $$

Again inserting $a = m + n$ leaves

$$ { \small \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, d}{2 \, \pi} \ \right] \, \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] = \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \ \right] + \text{Cosh} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right] \, \text{Sinh} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right]. } \tag{6} \label{eq:6} $$

Equations $(\ref{eq:5})$ and $(\ref{eq:6})$ are obviously equivalent, since $\text{Cosh}[ \, i \, z \, ] = \text{Cos}[ \, z \, ]$ and $\text{Sinh}[ \, i \, z \, ] = i \, \text{Sin}[ \, z \, ]$ for complex numbers $z$. (Of course, such equivalence also applies to begin with between equations $(\ref{eq:1})$ and $(\ref{eq:3})$, and between equations $(\ref{eq:2})$ and $(\ref{eq:4})$, resp.)

The generalization of Stewart's theorem to Lorentzian spaces, as a formula in terms of Lorentzian distances (of timelike triangles and their timelike or lightlike caveans), is therefore formally equal to its generalization to spherical geometry, as stated in the OP question.

Note on further generalization and a special case:
The derived formula $(\ref{eq:5}, \ref{eq:6})$ may not only apply to strictly timelike triangles (and their timelike or lightlike cevians) but for instance also to triangles which have two timelike and one lightlike side, say $c = 0$, whereby $\text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, c}{2 \, \pi} \ \right] = 0$.

Moreover specifying the cevian as lightlike as well, i.e. $d = 0$, equation $(\ref{eq:5})$ simplifies to

$$ \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, a}{2 \, \pi} \ \right] = \text{Cos} \! \! \left[ \, \frac{\sqrt{ k } \, b}{2 \, \pi} \ \right] \, \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, m}{2 \, \pi} \ \right] + \text{Sin} \! \! \left[ \, \frac{\sqrt{ k } \, n}{2 \, \pi} \ \right]. \tag{7} \label{eq:7} $$

Again inserting $a = m + n$ and expanding the trigonometric functions in terms of $\sqrt{ k } \ne 0$, terms up to second order cancel, and the terms of third order sum up to being proportional to

$$ b^2 = a \, n, $$

i.e. corresponding to a well-known relation in flat spacetime which in some instances seems to have been misattributed to "Robb, 1936", at least a version of which has however been presented by A. A. Robb, 1911, already.