🚀 go-pugleaf

Jürgen says...
>
>Daryl McCullough schrieb:
>
>>>>Right. c is only constant as measured in an inertial frame.
>>>
>>>At this point you should explain how to measure velocities (i.e. lengths 
>>>and durations) in non-inertial frames. Good luck!
>> 
>> Well, you can always define *coordinate* speed to be the
>> derivative dx^u/dx^0, the rate of change in the spacelike
>> coordinate x^u as a function of the timelike coordinate x^0.
>> If the coordinates x^u are not inertial Cartesian coordinates,
>> then there is no reason at all for this to be equal to c.
>
>Ok, that´s how you can *define* it. But the question was how to *measure*?

Yes, well in the case of noninertial coordinate systems, there *is*
no standard way of measuring the coordinates of an event (in inertial
coordinate systems, there is a standard way, using light-based
measurements). You have to look at the specific noninertial coordinate
system in order to say how measurements are done.

But for example, let's take the case of a rocket ship undergoing
constant acceleration (as measured by the passengers). A convenient
non-inertial coordinate system to use is this one:

    X = square-root(x^2 - c^2 t^2)
    T = c/g arctanh(ct/x)

where x and t are the inertial coordinates, and g is the acceleration
as measured in the center of the rocket. These coordinates can be
measured aboard the rocket simply enough:

    X = the distance along the direction of acceleration,
        as measured using onboard meter sticks. The zero of X
        is chosen so that the center of the rocket is at X = c^2/g
        where g is the measured acceleration at the center.

    T = c^2 tau/(X g)
        where tau is the time shown by an onboard clock that has always
        been at location X, synchronized so that tau=0 when the rocket
        first takes off. (T = tau only at the center of the rocket).

With these coordinates, the speed of light as measured aboard the
rocket is

    dX/dT = gX

So the coordinate speed of light is height-dependent.

--
Daryl McCullough
Ithaca, NY