Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.

If the trajectories of shrapnel particles a and b are separated by an
angle theta, what is the velocity of particle b relative to particle a?

At relativistically insignificant velocities the answer seems simple:

The tangential velocity Vt is v sin theta
The radial velocity Vr is v ( 1 - cos theta)
And the resultant vector magnitude is sqrt { Vr ^ 2 + Vt ^ 2}

= sqrt { [ v (1 - cos theta)] ^ 2 + [v sin theta] ^ 2}
= v sqrt { 1 - 2 cos theta + cos ^ 2 theta + sin ^ 2 theta }
= v sqrt { 2 - 2 cos theta}

= sqrt (2 Vr)

However, I am not so sure of the appropriate formulae when v is
relativistically significant.

Using the relativistic addition of velocities formula w = (u + v)/(1 +
uv/c^2), I still get:

Vt = v sin theta, but
Vr = v (1 - cos theta) / (1 - {[v/c] ^ 2} cos theta)

Thus, if v/c = s , I get
Vt = 0 when theta = 0, ~ sc/1.414 when theta = 45, and sc when theta =
90 degrees. Similarly,
Vr = 0 when theta = 0, ~ 0.414sc/(1.414 - s ^ 2) when theta = 45, and
sc when theta = 90 degrees.

These functions seem well behaved as s tends to 1 giving
Vt = 0 when theta = 0, ~ 0.707c when theta = 45, and c when theta = 90
degrees. Similarly,
Vr = 0 when theta = 0, ~ c when theta = 45, and still c when theta = 90
degrees.

Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

Chalky wrote:

Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.

If the trajectories of shrapnel particles a and b are separated by an
angle theta, what is the velocity of particle b relative to particle a?

At relativistically insignificant velocities the answer seems simple:

The tangential velocity Vt is v sin theta
The radial velocity Vr is v ( 1 - cos theta)
And the resultant vector magnitude is sqrt { Vr ^ 2 + Vt ^ 2}

= sqrt { [ v (1 - cos theta)] ^ 2 + [v sin theta] ^ 2}
= v sqrt { 1 - 2 cos theta + cos ^ 2 theta + sin ^ 2 theta }
= v sqrt { 2 - 2 cos theta}

= sqrt (2 Vr)

However, I am not so sure of the appropriate formulae when v is
relativistically significant.

Using the relativistic addition of velocities formula w = (u + v)/(1 +
uv/c^2), I still get:

Vt = v sin theta, but
Vr = v (1 - cos theta) / (1 - {[v/c] ^ 2} cos theta)

Thus, if v/c = s , I get
Vt = 0 when theta = 0, ~ sc/1.414 when theta = 45, and sc when theta =
90 degrees. Similarly,
Vr = 0 when theta = 0, ~ 0.414sc/(1.414 - s ^ 2) when theta = 45, and
sc when theta = 90 degrees.

These functions seem well behaved as s tends to 1 giving
Vt = 0 when theta = 0, ~ 0.707c when theta = 45, and c when theta = 90
degrees. Similarly,
Vr = 0 when theta = 0, ~ c when theta = 45, and still c when theta = 90
degrees.

Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr},

since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

Transformation of velocities parallel to the direction of motion.

(V1 + V2)/[1 +(V1)(V2)/c^2]

For velocities at an arbitrary angle theta, Jackson gives

u_parallel = (u'_parallel + v)/(1+(v dot u')/c^2)
u_perp = u'_perp/(gamma_v(1+(v dot u')/c^2))

Also,

http://www.physics.umanitoba.ca/~souther/waves02/feb0402/sld011.htm

Chalky wrote:

Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.

If the trajectories of shrapnel particles a and b are separated by an
angle theta, what is the velocity of particle b relative to particle a?

At relativistically insignificant velocities the answer seems simple:

The tangential velocity Vt is v sin theta
The radial velocity Vr is v ( 1 - cos theta)
And the resultant vector magnitude is sqrt { Vr ^ 2 + Vt ^ 2}

= sqrt { [ v (1 - cos theta)] ^ 2 + [v sin theta] ^ 2}
= v sqrt { 1 - 2 cos theta + cos ^ 2 theta + sin ^ 2 theta }
= v sqrt { 2 - 2 cos theta}

= sqrt (2 Vr)

However, I am not so sure of the appropriate formulae when v is
relativistically significant.

Using the relativistic addition of velocities formula w = (u + v)/(1 +
uv/c^2), I still get:

Vt = v sin theta, but
Vr = v (1 - cos theta) / (1 - {[v/c] ^ 2} cos theta)

Thus, if v/c = s , I get
Vt = 0 when theta = 0, ~ sc/1.414 when theta = 45, and sc when theta =
90 degrees. Similarly,
Vr = 0 when theta = 0, ~ 0.414sc/(1.414 - s ^ 2) when theta = 45, and
sc when theta = 90 degrees.

These functions seem well behaved as s tends to 1 giving
Vt = 0 when theta = 0, ~ 0.707c when theta = 45, and c when theta = 90
degrees. Similarly,
Vr = 0 when theta = 0, ~ c when theta = 45, and still c when theta = 90
degrees.

Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

Transformation of velocities parallel to the direction of motion.

(V1 + V2)/[1 +(V1)(V2)/c^2]

For velocities at an arbitrary angle theta, Jackson gives

u_parallel = (u'_parallel + v)/(1+(v dot u')/c^2)
u_perp = u'_perp/(gamma_v(1+(v dot u')/c^2))

Einstein derived it in full in 1905, but regretfully textbooks don't always
provide the full picture - see
http://www.fourmilab.ch/etexts/einstein/specrel/www/ , par.5.

Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.


Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

I don't know what vector you mean, particularly as you have an expanding

Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.

If the trajectories of shrapnel particles a and b are separated by an
angle theta, what is the velocity of particle b relative to particle a?

At relativistically insignificant velocities the answer seems simple:

The tangential velocity Vt is v sin theta
The radial velocity Vr is v ( 1 - cos theta)
And the resultant vector magnitude is sqrt { Vr ^ 2 + Vt ^ 2}

= sqrt { [ v (1 - cos theta)] ^ 2 + [v sin theta] ^ 2}
= v sqrt { 1 - 2 cos theta + cos ^ 2 theta + sin ^ 2 theta }
= v sqrt { 2 - 2 cos theta}

= sqrt (2 Vr)

However, I am not so sure of the appropriate formulae when v is
relativistically significant.

Using the relativistic addition of velocities formula w = (u + v)/(1 +
uv/c^2), I still get:

Vt = v sin theta, but

Vr = v (1 - cos theta) / (1 - {[v/c] ^ 2} cos theta)

Thus, if v/c = s , I get
Vt = 0 when theta = 0, ~ sc/1.414 when theta = 45, and sc when theta =
90 degrees. Similarly,
Vr = 0 when theta = 0, ~ 0.414sc/(1.414 - s ^ 2) when theta = 45, and
sc when theta = 90 degrees.

These functions seem well behaved as s tends to 1 giving
Vt = 0 when theta = 0, ~ 0.707c when theta = 45, and c when theta = 90
degrees. Similarly,
Vr = 0 when theta = 0, ~ c when theta = 45, and still c when theta = 90
degrees.

Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

Chalky

Thus spake Chalky <chalkyspam@bleachboys.co.uk
Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.


Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

I don't know what vector you mean, particularly as you have an expanding
sphere, so there is no particular direction for a vector.

The magnitude
of a velocity in sr is always 1, which you can find quite trivially.

Oh No wrote:

Thus spake Chalky <chalkyspam@bleachboys.co.uk
Consider an expanding sphere of shrapnel which propagates at speed v
from the source of an explosion.


Nevertheless, I am still stuck for what the total magnitude of the
vector is. Presumably it can't still be sqrt {2 Vr}, since that would
be greater than c for most of the angular distribution, when s tends to
1. Can anyone help out?

I don't know what vector you mean, particularly as you have an expanding
sphere, so there is no particular direction for a vector.

I thought I had made this clear. It is the vector velocity of any given
piece of shrapnel, relative to another.

Each will have a radial
component, relative to that reference piece, which is unambiguous. The
various pieces will also have a circularly symmetric angular
distribution, about the line joining our reference piece to the source
of the explosion.

The magnitude
of a velocity in sr is always 1, which you can find quite trivially.

This would seem to be either nonsense or meaningless.

It is obvious
that something with a velocity of zero relative to me, does not have
the same velocity, relative to me, as something with a velocity of c.
Perhaps you could explain your trivial method so that we all know what
to avoid in the future.

In sr you replace 3-vectors with 4-vectors. Usually the 0 direction is

In sr you replace 3-vectors with 4-vectors.

In article <6iLGZYtFcElFFwD8@charlesfrancis.wanadoo.co.uk>, Oh No
NotI@charlesfrancis.wanadoo.co.uk> wrote:

[snip]

In sr you replace 3-vectors with 4-vectors.

Yes, and 4-velocities are the simplest way to obtain the formulas the OP
was seeking, but the *relative* velocity between two pieces of shrapnel
is not a *unit* 4-vector, by any definition I can imagine.

It was pretty clear that the OP was seeking the relative speed between
two specified pieces of shrapnel, which does have an invariant meaning in
terms of the Lorentzian dot product between their 4-velocities:

harry wrote:

Einstein derived it in full in 1905, but regretfully textbooks don't
always
provide the full picture - see
http://www.fourmilab.ch/etexts/einstein/specrel/www/ , par.5.

It looks like Einstein didn't derive it in full, since his formula
gives highly superluminal velocities for exploding relativistic bombs
(such as a Hydrogen bomb detonated in the stratosphere).

At the classical limit, v<<c, the last term of the numerator vanishes,
and the denominator becomes 1. Einstein's equation thus reduces to:

V = sqrt (v ^ 2 + w ^ 2 + 2 v w cos a) = v sqrt 2 + 2 cos a

Now, when v = w, it is obvious that two bits of shapnel travelling in
the same direction at the same speed, will have zero relative velocity.
According to this formula, however, the velocity is 2 v. Clearly,
therefore, the classical limit formula should be

V = v sqrt (2 - 2 cos a)

In article <6iLGZYtFcElFFwD8@charlesfrancis.wanadoo.co.uk>, Oh No
NotI@charlesfrancis.wanadoo.co.uk> wrote:

[snip]

In sr you replace 3-vectors with 4-vectors.

Yes, and 4-velocities are the simplest way to obtain the formulas the OP
was seeking, but the *relative* velocity between two pieces of shrapnel
is not a *unit* 4-vector, by any definition I can imagine.

Greg Egan wrote:
In article <6iLGZYtFcElFFwD8@charlesfrancis.wanadoo.co.uk>, Oh No
NotI@charlesfrancis.wanadoo.co.uk> wrote:

[snip]

In sr you replace 3-vectors with 4-vectors.

Yes, and 4-velocities are the simplest way to obtain the formulas the OP
was seeking, but the *relative* velocity between two pieces of shrapnel
is not a *unit* 4-vector, by any definition I can imagine.

But all velocity four-vectors for objects moving at speed < c are unit
4-vectors. The 3-vector to 4-vector transition changes the velocity
vector to a unit direction four-vector. But this unit-ness is of
course using the Minkowski metric (dt/dtau)^2 - (dx/dtau)^2 = 1 where
tau is the proper time parameter.

But of course you can define any length for the non-null velocity
vectors. The spatial velocity will be the ratio of magnitudes of
spatial to time component of this vector. In short it is only the
direction of this vector in space-time which is meaningful and its
"slope" is the spatial velocity. This vector is the basis vector
corresponding to the time axis for the object in question. It is a
unit vector for the frame moving with the object (one second per proper
second) and so is a unit vector in all frames, (one second or
light-second per proper second).

I am a bit of a broken record on the subject but you can see the
analogy more strongly if you work within the matrix form (or vector
operator form) of the transformations and use hyperbolic trig.

Think of the analogy where one only sees the shadows of trees on the
ground (at noon) and from this defines the "slope" of the trees by the
spacing of its (uniformly spaced) branches' shadows. One can even
define a 2-vector to express the "vector slope" of the branch shadow
spacing. When one then looks at the trees instead of just their
shadows the direction in which they lean is expressible instead by a
unit vector in 3-space. We then prefer this vector or the angle at
which they lean for a magnitude. The "slope" is then tan(theta).

One can then relativize direction, speak of the direction of the trees
relative to any solar position and we have rotation matrices to
transform between frames.

So work out the problem using pseudo-rotation matrices to transform the
velocity of one particle into the frame of another. The velocities
involved will be hyperbolic tangents of the pseudo-angles
parameterizing the transformations. v = tanh(s) [in c=1 units such as
seconds + light-seconds = c meters].

Regards,
James

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.

Does nobody actually know what that answer is?