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Re: Relativity



Paul Camp wrote concerning my claim that a uniformly rotating disk also
uniformly translating along a direction contained in the plane of the disk
will be undergo the same Lorentz contraction of its shape as a non-rotating
disk translating in the same way.

Really? Can we show this? Because it isn't obvious to me. If I
imagine a small cube sitting on a rotating disk with no translational
velocity and allow the disk to rotate at relativistic speed, the cube
will be Lorentz contracted except when it is moving perpendicular to
my line of sight. Now add to that translational motion and I don't
immediately see why the rotational effect should go away.

Yes, really. An sketch of the derivation is below.
My claim for angular velocity independence of the Lorentz contraction is
for the circular outer edge of the disk. For this outer edge we write a pair
of parametric equations describing this circle in the (primed) frame in which
the center of the circle is at rest. We label each point on the rotating
circle with a value of theta (where 0 <= theta < 2*pi) the Cartesian
coordinates of these points are given as:
x' = R*cos(theta + omega*t') and y' = R*sin(theta + omega*t')

(Notice that these parametric equations can be combined to give the equation
of a circle of radius R, i.e. x'^2 + y'^2 = R^2, by elimination of the common
argument of the trig functions.) Now it is true that the radius of the disk
will be affected by the rotation process if the disk rotates very fast.
However, we take the value of R above to be the actual radius of the disk
under the rotating conditions.

Now we define a Lorentz boost along the x-direction with speed v.
In the new (unprimed frame) we have:
x = (x' + v*t')/sqrt(1 - (v/c)^2) , y = y' , and
t = (t' + v*x'/c^2)/sqrt(1 - (v/c)^2) where we have chosen the 4-origins of
the respective coordinate systems to coincide for convenience. Now the last
time equation above is inverted to give t' = (t - v*x/c^2)/sqrt(1 - (v/c)^2).
This inverted expression is substituted for the t' value in the arguments of
the trig functions in the parametric equations of the rotating circle. Next,
these time-substituted parametric equations are substituted in for the x' and
y' values in the LT equations which give x & y. The resulting parametric
equations give x and y as functions of t. The following equations result:

x = sqrt(1 - (v/c)^2)*R*cos(arg) + v*t, and y = R*sin(arg) where
arg = theta + omega*(t - v*x/c^2)/sqrt(1 - (v/c)^2) .

Notice that the equation for x is not really explicitly solved for x, but x is
given implicitly in this equation. Just like in the primed system we can
combine these parametric equations and eliminate the trig functions and their
common argument--arg. The result is:
((x - v*t)^2)/((1- (v/c)^2)*R^2) + (y^2)/R^2 = 1 .
This equation is the equation of an ellipse of semimajor axis R and semiminor
axis R*sqrt(1 - (v/c)^2) moving along the x direction with velocity v where
the minor axis is along the (x) direction of motion. The eccentricity of this
ellipse is just v/c. Notice that the equation for this ellipse is independent
of omega.

We can similarly find the motion for points on the interior of the circle by
repeating the above derivation for a sequence of concentric rings of ever
smaller radius. We see that each concentric circle is transformed to a
corresponding concentric ellipse of identical eccentricity and translational
velocity.

Q.E.D.

David Bowman
dbowman@gtc.georgetown.ky.us