# Relative velocity and lenticity

Consider a particle P in uniform motion. Suppose two inertial observers observe its motion. Observer K is stationary relative to the ground, and observer L is in uniform motion in the same direction as P but at a different rate.

(A) Say the spatial position of P relative to K is x(P, K), the spatial position of P relative to L is x(P, L), and the spatial position of L relative to K is x(L, K). The relation between these observations is

Differentiate these spatial positions with respect to time t to get time velocities (since time is the independent variable)

Differentiate these velocities v with respect to time to get accelerations

Since the acceleration of L relative to K is zero, the acceleration of the particle P measured by both inertial observers is equal.

(B) Now say the temporal position of P relative to K is t(P, K), the temporal position of P relative to L is t(P, L), and the temporal position of L relative to K is t(L, K). The relation between these observations is

Differentiate these temporal positions with respect to distance s to get space lenticities (since space is the independent variable)

Differentiate these lenticities w with respect to distance to get relentations

Since the relentation of L relative to K is zero, the relentation of the particle P measured by both inertial observers is equal. Note the symmetry with (A).

(C) Say the spatial position of P relative to K is x(P, K), the position of P relative to L is s(P, L), and the position of L relative to K is x(L, K). The relation between these observations is

Differentiate time t with respect to these spatial positions to get time lenticities (since time is the independent variable)

with the boxplus representing reciprocal addition. Why not addition? Because, apart from division by zero,

Differentiate time with respect to these lenticities w to get relentations

Since the relentation of L relative to K is infinite, , the relentation of the particle P measured by both inertial observers is equal.

(D) Say the temporal position of P relative to K is t(P, K), the temporal position of P relative to L is t(P, L), and the temporal position of L relative to K is t(L, K). The relation between these observations is

Differentiate distance s with respect to these temporal positions to get space velocities (since space is the independent variable)

with the boxplus representing reciprocal addition. Why not addition? Because, apart from division by zero,

Differentiate distance with respect to these velocities v to get accelerations

Since the acceleration of L relative to K is infinite, the acceleration of the particle P measured by both inertial observers is equal. Note the symmetry with (C).