Re: Doppler effect: Reflection of light from a moving mirror
- From: Albertito <albertito1992@xxxxxxxxx>
- Date: Tue, 17 Feb 2009 02:10:46 -0800 (PST)
On Feb 16, 7:55 pm, PD <TheDraperFam...@xxxxxxxxx> wrote:
On Feb 15, 7:00 am, Albertito <albertito1...@xxxxxxxxx> wrote:
A source of light and an observer are at rest in the same
inertial frame, where r is the vector distance between them.
A mirror, moving at receding velocity u perpendicular to r
through its midpoint, at height h, reflects light of source
off to observer.
We can know the component velocities v_1 and v_2, of velocity
u, with respect to source and to observer, respectively. Those
component velocities actually have the same magnitude, v,
so we can write
u = v_1 + v_2,
Are you sure this right?
Consider the case where |h| = |r| / 2, so that at that moment, the
mirror, the source, and the observer form a right isosceles triangle.
In this case, the velocity u as you've defined it is NOT the vector
(or algebraic) sum of the velocity between the mirror and the source
and the mirror and the observer. You'd be off by a factor of two. And,
if you'll look a little further down, you'll see that factor of 2 in
your own equations.
v = |v_1| = |v_2|,
Well, you are right. In the setup, I assumed u is
a velocity constant in magnitude, perpendicular to
r in its midpoint. So, the mirror is receding with
respect to source at a velocity v_1 whose magnitude
is
|v_1| = v = |u|/sqrt[1 + (|r|/2|h|)^2],
and wrt observer it is receding with a velocity v_2
exactly with the same magnitude v. Amazingly, the
magnitude v is not constant if |u| is constant, and
that means the mirror is receding from source and
observer at an accelerated motion, which goes from
v=0 at |h|=0, to v=|u| at |h|=oo.
Fortunately, that "error" of mine is irrelevant for
the subsequent steps of the rationale.
and the angle alpha between v_1 and v_2 as
alpha = 2*arctan(|r|/2h)
so,
|u| = 2*cos(alpha/2)*v,
v = |u|/[2*cos(alpha/2)].
The observer then detects the reflected light as if it were
coming from the image behind the mirror. Since the mirror
is moving with velocity v_1 wrt to source and v_2 wrt observer,
it creates a virtual image of the source moving with receding
velocity w = 2v along the observer line of sight,
w = 2v = |u|/cos(alpha/2)
No, this is too broad a statement. The *image* is receding from the
observer at this speed, but this does not mean that one can expect a
Doppler shift according to this speed. This is a simple case of
overextending *some* properties of this optical case to apply to *all*
properties, an inappropriate extrapolation.
Nature doesn't bother about what one can expect. Nature doesn't
bother about SR being wrong or right. There is no model that could
exactly conform to nature whatsoever.
Then, for that virtual velocity w, which can even be a
superluminal velocity, because it doesn't correspond to
any real motion between source and observer (recall source
and observer are really at rest),
Yes, this last comment of yours is directly pertinent to my previous
remark.
Don't use the pertinence as a bias supporting your arguments.
What you have to do is to compute correctly the predicted
Doppler under SR, in the above setup, and see whether SR
predicts any Doppler or not.
we can predict an observed
Doppler frequency red-shift of the original frequency f_0
emitted by source, as
f = f_0*Exp[-w/c] = f_0*Exp[-|u|/(c*cos(alpha/2))].
Regards
*Appendix:
Under SR, the prediction is as follows.
Apply an Einstein's addition of velocities,
w = 2v/(1 + v^2/c^2),
Now, apply a relativistic Doppler,
f' = f_0*sqrt[(1 - w/c)/(1 + w/c)],
and after some algebra, knowing that v = |u|/[2*cos(alpha/2)],
it yields
f' = f_0*[(|u| - 2*c*cos(alpha/2)))/
(|u| + 2*c*cos(alpha/2)))]
Where is the misleading mistake in this SR derivation?
There are two misconceptions that when acting cooperatively
try to slightly compensate the wrong answer to a right one.
The first error is to assume there must exist a relativistic
addition of velocities as w = 2v/(1 + v^2/c^2). This is
nonsense, since w is a VIRTUAL velocity of source wrt observer
(they are actually at rest), not a real one (it can be even
virtually superluminal), a relativistic addition of velocities
must NOT be applied. If the virtual w is superluminal it
means that, once the observer detects the reflected light
as if it were coming from the image behind the mirror, the
information is not FTL, because that information has travelled
a larger path length than that of a straight line from source
to observer, so actually that information travelled with the
light speed, it is not FTL. This error of misconception is
slightly corrected/compensated when the SR's relativistic
Doppler formula is applied to f_0 through that wrong w,
yielding a predicted red-shifted frequency f' that is very
close to the correct one f, provided above. Actually, f/f_0
and f'/f_0 only start to differ from the third order term of
their respective Taylor expansion series,
f/f_0 = 1 - |u|/(c*cos(alpha/2)) +
|u^2|/(2*c^2*cos^2(alpha/2)) -
|u^3|/(6*c^3*cos^3(alpha/2)) + ...
f'/f_0 = 1 - |u|/(c*cos(alpha/2)) +
|u^2|/(2*c^2*cos^2(alpha/2)) -
|u^3|/(4*c^3*cos^3(alpha/2)) + ...
.
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