A perhaps simple way to understand the relativity of time and velocity
Three Larrikins, whom we may
name April, Mac and John decided once upon a time to test a fundamental premise of Special Relativity and the measurement
of time.
They hired a chaffeur-driven limousine of say 4 meters in length and defined a coordinate system for
that limousine centred at its midpoint (x,y)=(0,0). The limousine would travel due East with Mac positioned at a coordinate
1 meter to the West at (x,y)=(-1,0) facing towards the East. John would face to the West, opposite Mac at a coordinate
(+1,0). Mac and John would then remain stationary for the duration of the experiment and precisely 2 meters apart with a so
equidistant lightsource at the centre.
The limousine would travel at a speed v, say 80 km/h from the West and past April,
positioned due South at a coordinate (0,-3).
The larrikins now decide to label the event A to be the precise
instant for the centre of the limousine to be coincident with the y=0 coordinate for the stationary April. Precisely
at that instant,
the lightsource would switch on and send an omnidirectional lightsignal to the measuring instruments
of the three larrikins.
Both, Mac and John now define the event B to be the reception of the signal. As
both Mac and John are stationary relative to each other the lightpath x=ct will be identical, if the postulates of Special
Relativity hold. In particular, the time measured should be the same as tMac=tJohn=x/c=1/c
seconds. Their positional coordinates also remain unchanged at xMac=-1 and xJohn=+1
relative to their stationary frame of reference within the limousine.
But the situation is not as clear for April.
She sees the limousine travelling past her at speed v; which changes the lightpath. April actually postulates, that she will
observe Mac to receive the lightsignal before John; because the light must travel further to get to John's photo receptor,
than to Mac's.
So April measures two events B; B1Mac and B2John. The
lightpath becomes ambiguous in x'John=(c+v)t'John and x'Mac=(c-v)t'Mac and due to a simple Galilean coordinate transformation or 'shifting of the x-axis' and for an apparent 'addition
of the velocities'.
April postulates a first approximation for her measurement of the B event in deriving t'John=x'John/(c+v)=x'John(1/c)/(1+v/c)=x'JohntJohn/(1+v/c) and t'Mac=x'Mac/(c-v)=x'Mac(1/c)/(1-v/c)=x'MactMac/(1-v/c).
April then takes the geometric
mean of her B events to get a second approximation in:
tApril=√(t'John.t'Mac)=√{x'John.x'Mac(1/c²)/(1-(v/c)²)}.
But (1/c²)=(tJohn.tMac)/(xJohn.xMac) as the lightpath relative to the limousine and with tJohn=tMac
for the expression:
tApril=tMac.√[x'Mac.x'John/xMac.xJohn]/√(1-(v/c)²).
April's
second approximation so begins to crystallise relativistic time dilation in proper time tMac=tJohn and c-invariance xMac=x'Mac and xJohn=x'John for relative reference frames.
The xB1Mac coordinate becomes xApril1=(-1+v/c)/√(1-(v/c)²)
and the xB12John coordinate becomes xApril2=(+1+v/c)/√(1-(v/c)²)
as the Lorentz-Coordinate transformation for displacement.
This is generalised as: X'=(X-VT)/√(1-(V/c)²)
for two reference frames O and O' moving relative to each other at a speed V.
Also the Lorentz-Contraction appears
in the proper length being xApril and the contracted length becoming ±1+v/c relative the
two Aprillean measurements.
The Lorentzian time coordinates transform similarly in adding a v.x/c² term
to the proper time in the time dilation expression.
T'=(T-VX/c²)/√(1-(V/c)²).
April's
precise measurement, so would give the results:
tApril1=(1/c-v/c²)/√(1-(v/c)²)=(1/c)√{(1-v/c)/(1+v/c)}
and tApril2=(1/c+v/c²)/√(1-(v/c)²)=(1/c)√{(1+v/c)/(1-v/c)}.
One
can now see, that tApril1=tMac√{(1-v/c)/(1+v/c)}< tMac
and that tApril2=tMac√{(1+v/c)/(1-v/c)}> tMac for all velocities v<c and tMac=tJohn. As the second measurement
gives the relative time measurement for the lightpath with respect to John however; April observes the time taken for the
light signal to reach John as being longer than that for travelling to Mac by the time differential:
tApril2-tApril1= tMac{√[(1+v/c)/(1-v/c)]-√[(1-v/c)/(1+v/c)]}.
In
our example of a speed of 80 km/h or 22.22..m/s; this time difference calculates as about (1.000000074..- 0.999999925..)/c~
0.00000015/c seconds or so 0.5 femtoseconds (10^-15).
One can obtain the time dilation
formulation by simple geometry in postulating April to receive the lightsignal at the instant the coordinates for the moving
limousine to coincide as the origin in x=y=0.
This means, that the lightpath is 'pythagorised' in Mac sending
a signal when the limousine's origin is at x=-vt and John receives Mac's signal for the origin to have moved to x=+vt.
A
stationary limousine so would define a lightpath for April of 2y=ct (or 6 meters in our example), being the return trip of
the signal from the source to April and back again. Here t is a proper time, measured in a stationary reference frame.
The
moving limousine vectorises as y and vt' with t' here being the improper time for a lightpath of [½ct']²=[½ct]²+[½vt']².
Solving
for t' gives: t'=√{(c²t²/(c²-v²)}=t/√{1-(v/c)²} and which is the time dilation formula
of Special Relativity.
Relative Velocity
Much confusion exists for the general reader, when attempting to apply Special Relativity to velocity concepts.
Using the above we shall here try to simplify the scenario in considering three cases.
Case A: A rocket moves away from the earth at relativistic speed and fires a projectile also at relativistic speed,
relative to itself. What is the projectiles speed relative to the earth?
Case B: Two rockets approach each other and collide at relativistic speeds as measured on the earth. What is the
relative speed of the centre of mass of the system or what is the velocity of the first rocket relative to the second rocket?
Case C: Two rockets travel parallel at relativistic speeds as observed on the earth. What is the relativistic speed
of the centre of mass?
Generally, the Lorentz transforms gave the coordinate transformas for displacement and for time. One now differentiates
dX'/dT'=U'x in the differentials or one simply defines:
U'x= X'/T'=[(X-VT)/√(1-(V/c)²)]/[(T-VX/c²)/√(1-(V/c)²)].
Dividing the expression by T in numerator and denominator gives:
U'x=[X-VT]/[T-VX/c²]=[Ux-V]/[1-VUx/c²].
Here U'x depicts the velocity in reference frame O' and Ux represents the velocity in frame O with V the velocity relative
to the moving frames. The velocity V can be said to be a relative velocity of a third 'particle', measured in the two reference
frames. V is positive in the the +x direction and negative in the opposite.
If we use the inverse Lorentz transformation (exchanging indices and replacing V by -V), then the symmetry of the postulates
of Special Relativity gives Ux=[U'x+V]/[1+VU'x/c²]. (Solving algebraically gives of course the same result).
It is advisable to use this formulation in cases such as Case A, where the moving frame relative to earth superposes another
moving frame.
Case A:
We choose the rocket to move at 0.8c relative to the earth and emitting a projectile in the forward motion at 0.4c.
In this example U'x=0.8c in frame O' and as the projectile moves relative to this frame at 0.4c, the frames move relative
to each other at this speed V=0.4c. We seek the relative velocity of the 'particle' relative to the other frame O that is
velocity Ux.
Galilean Relativity would say the projectile moves at 1.2c relative to the earth; but our expression above cannot
ignore the denominator and we get:
U'x=[1.2c]/[1+0.32]=0.9091c.
No relative velocity can ever exceed the speed of light.
Case B:
We choose a rocket A moving to the right at 0.8c and the rocket B moving to the left at 0.6c as measured in the
earthframe O. This situation must also add the velocities as in the classical Galilean case for collinear collisions.
We seek the velocity of rocket A relative to frame O', labelling rocket A as the 'particle' with velocity Ux relative
to the earth.
Then Ux=0.8c and V=-0.6c in frame O for U'x=1.4c/[1.48]=0.9459c as the velocity of the centre of mass or as the velocity
of rocket A as measured by rocket B in its reference frame O' (relative to itself at rest).
Reassigning the frames in seeking the relative velocity of rocket B relative to rocket A gives the same result in a varied
numeracy. Here Ux=0.8c in frame O of rocket A and
U'x=-0.6c=[0.8c-V]/[1-0.8V/c] in frame B and the earth observer becomes the 'particle' with relative frame velocity V.
Solving for V=1.4c/1.48 via -0.6c+0.48V=0.8c-V gives the previous result, now for V.
Case C:
This case simply subtracts the velocities. We choose rocket A to move right at 0.8c and superpose rocket B
moving also to the right at 0.6c. A classical Galilean transform would predict a relative speed of 0.2c between the rockets.
Let Ux=0.8c in O and seek U'x in O' i.e the velocity of rocket A relative to rocket B.
U'x=[0.8c-0.6c]/[1-0.48]=0.3846c.
Also, 0.6c=[0.8c-V]/[1-0.8V/c] for 0.6c-0.48V=0.8c-V for V=0.2c/0.52=0.3846.
Rocket A moves to the right at 0.3846c relative to rocket B and rocket B moves to the left at 0.3846c relative to rocket
A and with the centre of mass moving at 0.3846c towards the right.
Tony B.
Hi April!
The X' refers to coordinate systems and not to any calculus. There is no calculus in this post,
except as basic differential and the difference between two measurements.
Two events, say the sending and
receiving of a signal are measured in TWO reference systems O and O' say with reference frame O moving with velocity v relative
to reference frame O' or vice versa.
Much confusion then arises in failing to properly define the coordinates
in a manner of 'who moves relative to what'.
So, a 'proper time' can be defined in either sysyem; 'proper' meaning
that the events coincide in whatever reference frame thus defined - they have the same locality and the measurements are taken
at the same place. Then relative to that frame an 'improper' time results in the quandaries (or illogicalities wrt common
sense) of Special Relativity. The 'improper' time is measured at different places.
So in the example
both April's time and Mac's time can be the 'proper' time and be 'dilated' relative to the other under the coordinate definitions.
But
this is used technically to define instantenuity and simultaneity in the Lorentz formulas, which I tried to derive from a
more simple premise, using the geometric mean of differentials.
Tony B.
The Rising of the Sun, The Running of the Deer
The holly and the ivy,
When they are both full grown,
Of all trees that are in the wood,
The holly bears
the crown
Yuletide Carol from Druidic Origins
Blessings and Joy on the Return of the Light
(posted by Jordan Stratford+, 21.12.2004)
