Inertial reference frame and Bondi k-calculus

[cianfa72]

TL;DR Summary

About the definition of standard coordinate time in SR context compared to Bondi k-calculus

Hi, I was reading the Bondi k-calculus as introduced in R. d'Inverno book and Bondi k-calculus.
As far as I understand, in the context of SR, the radar time coordinate from an inertial observer/clock is basically the same as the coordinate time $t$ assigned to any event from an inertial frame in which the coordinate clocks at rest in it have been Einstein's synchronized with that inertial "master" clock.

Is that the case ? Thanks.

 

[Dale]

Yes, radar coordinates reduce to a standard inertial frame for an inertial radar.

 

[cianfa72]

Yes, radar coordinates reduce to a standard inertial frame for an inertial radar.

Ok, so in 3D space one can "recover" the event's standard spatial coordinates $x,y,z$ (w.r.t. the radar location) from the event's distance $d = \frac c 2 \left (T_2 - T_1 \right )$ and the direction the radar echo comes from ($T_1$ and $T_2$ are proper times along the inertial radar worldline).

 

[cianfa72]

A question somewhat related to the topic. By definition of inertial frame (aka inertial chart) the spacetime path of free objects (zero proper acceleration as measured by accelerometers attached to them) are described in such a chart by paths with zero coordinate acceleration.

What about the spacetime path of objects at rest in a such inertial chart ? My reasoning is as follows:

Consider the spacetime timelike path described by fixed spatial coordinates $(x,y,z)$ in that inertial chart -- call it $\alpha (\tau)$. At any point in spacetime there is unique geodesic in a given spacetime direction (therefore in particular there is a unique timelike geodesic in a given timelike direction). Take the timelike geodesic tangent to timelike path $\alpha$ at event $P$ -- call it $\beta(\tau)$. It is at rest in the inertial frame at $P$ therefore it must coincide with $\alpha$ everywhere (otherwise $\beta$ would not have zero coordinate acceleration at $P$).

 

[PeterDonis]

What about the spacetime path of objects at rest in a such inertial chart ?

The definition of "inertial chart" already gives you the answer to this question. Your roundabout "reasoning" implicitly makes use of the fact about inertial charts that answers the question, so you actually aren't proving anything, you're just laboriously restating the definition.

 

[cianfa72]

The definition of "inertial chart" already gives you the answer to this question. Your roundabout "reasoning" implicitly makes use of the fact about inertial charts that answers the question

My definition of inertial frame/chart: a reference frame/chart such that any free body worldline (i.e. with zero proper acceleration) has zero coordinate acceleration in it.

The above definition does not require/imply, at first glance, that a body at rest in it must have zero proper acceleration.

 

[PeterDonis]

My definition of inertial frame/chart

Your definition is irrelevant. What matters is the standard definition in the literature.

 

[PeterDonis]

The OP question has been answered. Thread closed.