On Jan 18, 12:32 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
Greg Egan wrote:
In article <1168883430_1...@sicinfo3.epfl.ch>, harry
harald.vanlintelButNotT...@epfl.ch> wrote:
"Chalky" <chalkys...@bleachboys.co.uk> wrote in message
news:1168758870.922068.265170@51g2000cwl.googlegroups.com...
[snip]
All I want to know are the velocities of
all other schrapnel particles relative to any given one, as a function
of the dispersion angle theta of their trajectories, as measured in a
reference frame which is stationary relative to the source of the
explosion.
[snip]
It's strikingly simple. This relative velocity is sometimes called "closing
velocity", in order to stress that this is *not* the relative velocity of
frame transformations. You don't switch frames; the rules of vector addition
are the same for high speeds as for low speeds!
This should not be confused with "Galilean" versus "relativistic" speed
addition which happens in frame transformations. The confusion sometimes
arises because in classical mechanics the two are mathematically the
same -not so in relativistic physics. See also how Einstein used standard
vector mathematics inside the same paragraph 5 in which he derived the
relativisitc addition equation: there he mentions that the relative speed of
a moving point and a light ray in the original reference frame is simply
(c-v).
Harald
I parsed the OP's question very differently! Although it's clear that
he's measuring *theta* in the centre-of-mass frame of the explosion, I
thought he wanted the velocity of one piece of shrapnel *as measured in
the frame of another piece of shrapnel*.
That is correct
Of course if he wants everything as measured in the centre-of-mass frame,
the whole problem's trivial, and he can subtract the 3-vectors in that
frame.
No. That is not what I wanted
But the fact that he used the relativistic addition of velocities formula
in his original post made me think he was after an answer in the shrapnel
frame.
Absolutely correct
To be clear, if he does what you're suggesting then he'll get
relative speeds of up to 2v for oppositely directed shrapnel. This is
the old "one spaceship goes at 0.9c to my left, and another goes at 0.9 c
to my right, so I see them separating at 1.8c". Which is perfectly true,
and if that really is what the OP wants, then he doesn't need relativity
at all.
If that is true, I will not bother paying any further attention to
Harald's post.
Chalky- Hide quoted text -
- Show quoted text -
I have been giving this subject some further thought recently, given
that Chalky originally asked the SR question while some of us were
still struggling to define a simple inertial reference model of an
observable expanding universe, relative to which accelerating
expansion could be plotted (during a still eariler discussion over at
sci.astro.research). In practice, the Milne (empty) model of EFE is
used for that purpose by astronomers, and this can be tabulated for z
shift as a function of distance, simply by plugging Omega_Lamda = 0
and Omega_M =0 into, for example, Ned Wright's Cosmic calculator.
Admittedly, Chalky originally asked about z shift as a function of
dispersion angle, but this question could equally well have been asked
as a function of physical separation (or luminosity distance).
Now, if I understand the foundations of the theory correctly, when
Lambda and M are both set to zero, there should be no gravity and GR
thus approximates to SR, giving an inertial model of both the
expanding universe and, presumably, of exploding relativistic bombs
(when gravity is negligible), as the schrapnel velocity approaches c.
However, since I have arrived at this conclusion from a philosophical
perspective as opposed to rigorous mathematical analysis, informed
comments from others would be appreciated.
