# How distance is determined between two inertial frames in empty space using Bondi’s radar method?

I have read how Bondi’s radar method can be used between two vessels in empty space to determine their relative distance and also if their distance is increasing or decreasing. Can Bondi's radar method be used to determine their speed and few other details with respect to distance.

For instance the vessels distance can increase by both moving in opposite direction with equal speed plus it can also be that one has greater speed than the other. Or if both are moving in the same direction with the vessel ahead having greater speed.

If both vessels are moving in the same direction, the one ahead will get redshifted and also it would take longer for signal to reach it since it is moving away from the signal but the opposite is true for the vessel pursuing it. It sees blueshift and since it is moving towards the signal it also takes less time for the signal to arrive for it compare to the vessel ahead. But round trip of the signal would increase in time for both in their own frame.

How can each vessel determine if the other is redshifted or blueshifted?

For the vessel ahead that is redshifted might assume that the other vessel is moving away or moving in the same direction but at lower speed.

For the vessel behind noticing distance increase while at the same time blueshifted would know that it is moving slower but the same direction if I have the doppler effects right.

Thank you very much.

Using the radar method of the Bondi k-calculus (e.g., my article https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/ as suggested by @trula), one records the clock-readings when the observer sends out a signal and receives its echo from scattering off a distant event.

• For simplicity, assume the worldlines of inertial astronauts Alice and Bob are coplanar. So, if their worldlines are parallel, they never meet and would presumably have a constant distance separating them. Otherwise, they met or will meet momentarily at a common event O, and thus have a nonzero relative-velocity.

• When Alice's watch reads $$t_{1s}$$, she sends out a signal and receives an echo at $$t_{1r}$$, reflecting off a distant event $$P$$ on Bob's inertial worldline. For $$P$$, Alice assigns a time coordinate $$t_P=(t_{1r}+t_{1s})/2\qquad \mbox{[the halftime]}$$ and position coordinate [divided by c] $$x_P/c=(t_{1r}-t_{1s})/2\qquad\mbox{[half the round-trip time]}$$

• At a later time, she sends another signal and records the watch readings, and assigns coordinates to another event $$Q$$ on Bob's inertial worldline. $$t_Q=(t_{2r}+t_{2s})/2\qquad x_Q/c=(t_{2r}-t_{2s})/2.$$

Alice can then define the spatial and temporal displacements $$\Delta x_{PQ}=x_Q-x_P$$ $$\Delta t_{PQ}=t_Q-t_P$$ and the relative velocity of Bob, according to Alice, $$v_{Bob\ wrt\ Alice}=\frac{\Delta x_{PQ}}{\Delta t_{PQ}},$$ which can all be written in terms of the watch readings $$t_{1s}$$, $$t_{1r}$$, $$t_{2s}$$, and $$t_{2r}$$.

• If $$\Delta x_{PQ}=0$$, then $$x_Q=x_P$$ (that is, $$(t_{2r}-t_{2s})/2=(t_{1r}-t_{1s})/2$$), the inertial astronauts are at rest with respect to each other. Expressed another way, we have $$\Delta t_r \equiv (t_{2r}-t_{1r}) \stackrel{relative\ rest}{=} (t_{2s}-t_{1s}) \equiv \Delta t_s,$$ where we defined have the elapsed times between receptions (the period-between-receptions) and between transmissions (the period-between-transmissions).

• If $$\Delta x_{PQ}>0$$, then $$x_Q>x_P$$ (that is, $$(t_{2r}-t_{2s})/2>(t_{1r}-t_{1s})/2$$), the inertial astronauts are separating. Expressed another way, we have $$\Delta t_r \stackrel{separating}{>}\Delta t_s.$$ We write this as $$\Delta t_r =k^2 \Delta t_s,\qquad \mbox{where k>1 for separating [redshift]}.$$

• Similarly, $$\Delta t_r =k^2 \Delta t_s,\qquad \mbox{where 0

• The relative-rest case can be expressed as $$\Delta t_r =k^2 \Delta t_s,\qquad \mbox{where k=1 for relative-rest [no shift]}.$$

(The $$k$$-factor is squared because there are two Doppler factors in this process.)

With some algebra, it can be shown that $$k_{Bob\ wrt\ Alice}=\sqrt{\frac{1+v_{Bob\ wrt\ Alice}}{1-v_{Bob\ wrt\ Alice}}}$$ is the Doppler factor (the Bondi k-factor). The Relativity Principle will imply that $$k_{Bob\ wrt\ Alice}=k_{Alice\ wrt\ Bob}$$ so just call each side $$k$$ for simplicity.
And $$k$$ can be expressed in terms of the four watch readings $$t_{1s}$$, $$t_{1r}$$, $$t_{2s}$$, and $$t_{2r}$$.

One could introduce a third inertial worldline to study the other two. It is simplest if this inertial observer measures the other observers moving in the same direction [or else you'll have to manage all of the minus sign differences that arise].

In addition, one can then derive the equations for the Lorentz boost and for the velocity-composition law.

since both ships move at constant speed it is easier to take one as the reference system which is not moving, than just draw the appropriate diagrams with Bondi's k factor and look at the results. maybe look at https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/