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Re: [Phys-L] another DIY relativity experiment



Regarding Bob S's most recent question:

David,
Please let me tax your patience with this stupidity which is
boggling my mind:
The analysis says that to an accelerating space ship there is a
stopped photon at z = -c^2/g. To the inertial observer K' this
is a photon moving at z' = ct' - c^2/g.
Did the space ship's acceleration bring this photon into
existence, or would the inertial observer have experienced this
photon even without the accelerating space ship?

The actual existence of the photon is not necessary at all. Its existence or non-existence has nothing to do with how either frame is put together. The concept only came up because I was describing the properties of the singular coordinate horizon in frame K @ z = -(c^2)/g. I pointed out that if any object is at rest there in frame K it experiences no proper time of its own, and it is moving at speed c in frame K'. The only possible kind of object such a hypothetical object could possibly be is a mass-less one, such as a photon, for instance. Its existence in the first place is purely hypothetical for the illustrative purpose of describing the kind of motion that occurs in K' for any object at rest at the horizon that only exists as a horizon in frame K. The issue of the existence or not of the mass-less object/photon is a matter of the physics of the situation. The physics is what it is *objectively* happening regardless of any observer's frame. But different observers at rest in different frames will tend to experience the objective physics of the situation in a more *subjective* frame-dependent way. It just so happens for the situation described above an observer at rest in K' experiences the photon always moving at constant speed c. But an observer at rest in the non-inertial K frame experiences the same mass-less object at the horizon just sitting still hovering there. This is similar to the situation of an outbound photon sitting on the event horizon of a a black hole as understood and described by an external observer at rest w.r.t. the hole. In fact this hovering-of-mass-less-null-object property can be thought as a *defining* property of a horizon. On one side of the horizon a mass-less object can reach the observer, and on the other side of the horizon (if the other side actually exists) it cannot. On the boundary between these two situations the mass-less object just hovers there never making any actual progress toward the observer.

To see how a freely falling object behaves in both the K' and K frame first consider any (positive mass) freely falling object moving along the z direction in frame K' with velocity v_0. If the object is located at position z' = z_0 at time t' = 0. the trajectory of the object is the simple motion:

z'(t') = z_0 + v_0*t'.

This object (according to N1) has the constant velocity v = v_0.

In the accelerating spaceship's K frame the same object has the trajectory:

z(t) = (z_0 + (c^2)/g)/(cosh(g*t/c) - (v_0/c)*sinh(g*t/c)) - (c^2)/g.

The coordinate velocity v = dz/dt for this motion is:

v(t) = (1 + g*z_0/c^2)*(v_0*cosh(g*t/c) - c*sinh(g*t/c))/(cosh(g*t/c) - (v_0/c)*sinh(g*t/c))^2

In the special case situation where v_0 = 0 so the object is initially at rest in K and is always at rest in K' these equations boil down to:

z(t) = (z_0 + (c^2)/g)/cosh(g*t/c) - (c^2)/g &

v(t) = -c*(1 + g*z_0/c^2)*tanh(g*t/c)*sech(g*t/c).

In the opposite limit, if the object is mass-less, and therefore necessarily moving at speed c (v_0 = ± c along the z direction) in inertial frame K' these equations boil down to:

z'(t') = z_0 ± c*t',

z(t) = (z_0 + (c^2)/g)*exp(±g*t/c) - (c^2)/g &

v(t) = ±c*(1 + g*z_0/c^2)*exp(±g*t/c).

Notice that whenever the object's initial location is not at the horizon (z_0 ≠ - (c^2)/g) then the object eventually approaches it as the time t approaches future infinity in frame K. This is always true no matter what the initial velocity of the object is. OTOH, if the object's initial location is already at the horizon, then its stays there at rest (in K) for all time t.

In all the above cases in frame K the value of v(t) above is the *coordinate* velocity in K. This is not necessarily the actual local physical velocity of the object (except when the object happens to be at location z = 0 when v is measured). This is because the coordinate velocity is a derivative of position w.r.t. the coordinate time, t rather than w.r.t. the local physical proper time (and in our particular case that coordinate time happens to be only the time kept by a clock at rest at z = 0). If the velocity v is measured using the time kept by a clock attached to the local starting position of the object at z = z_0 where it was released/launched at initial velocity v_0 (but that clock is attached to the spaceship rather than to the released object that falls away from the spaceship after it is released) then the velocity instead is:

v = (v_0*cosh(g*t/c) - c*sinh(g*t/c))/(cosh(g*t/c) - (v_0/c)*sinh(g*t/c))^2

in the generic initial velocity v_0 case. In the special case of the object released from rest, i.e. v_0 = 0, then this velocity is:

v(t) = - c*tanh(g*t/c)*sech(g*t/c).

In the opposite mass-less v_0 = ± c case this velocity (i.e. using the initial local clock in K) is:

v(t) = ±c*exp(±g*t/c).

Next now imagine on board the spaceship there is a sequence of very many clocks/observers, each one at rest in K, along the path of the object with a different clock for each local value of height z. As the object passes by each of these observers/clocks the object's local velocity is measured by that local observer. IOW, suppose the velocity is measured at each point of the object's trajectory using the local clock attached to the spaceship it happens to be passing by at that very moment. In such a situation when the initial velocity of the object is v_0 the always-locally measured velocity in K is:

v = (v_0*cosh(g*t/c) - c*sinh(g*t/c))/(cosh(g*t/c) - (v_0/c)*sinh(g*t/c)).

In the special v_0 = 0 case this always-local velocity is:

v = -c*tanh(g*t/c).

In the mass-less opposite v_0 = ± c case this always-local velocity is:

v = ±c.

Thus we see that, even in an accelerated frame, a mass-less object always *locally* passes by any observer at exactly speed c. This is the general situation, any massive object will locally pass by any observer at a speed strictly less than, c and a mass-less object will always locally pass by any observer *at* speed c. But it is only for the inertial frames of SR where for these speed limit rules this locality stipulation can be relaxed to even non-locally measured or velocities inferred from a distance using a distant clock. In general accelerated frames or in situations with (even more general) true gravitation with nonzero space-time curvature the locality stipulation on the speed limit rules must be retained if the speed limit rules are to be enforced. In the general gravitating and accelerated cases there are no a priori speed limit rules on the speed of objects (even on time-like paths) that are non-locally measured, or inferred from a distance using a clock distant from the moving object in question. The reason for this simply is that in accelerated frames, or in frames with true gravitation with nonzero space-time curvature, spatially separated clocks at rest w.r.t. each other may well not keep common time, and using clocks that keep time differently in different places makes for different velocities when derivatives w.r.t. these different mutually dilated time scales are used in calculating the 'velocities'.

David Bowman