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Lorentz Transformation |
The Lorentz Transformation is the very heart of the relativistic view of reality. It transforms the coordinates of an event in one inertial frame of reference into the coordinates of the same event in another frame. By applying this transformation to every event of interest, it is possible to transform the whole time-varying three-dimensional world perceived by one observer into the time-varying three-dimensional world perceived by another observer. In other words, the Lorentz transformation allows observers to switch back and forth between reference frames to see how relativistic quantities appear differently to different observers.
Hendrik A. Lorentz first derived the Lorentz transformations in 1890, some fifteen years before Einstein introduced the theory of special relativity. Lorentz deduced the transformations from the properties of electromagnetic radiation in an effort to explain the null result of the Michelson-Morley experiment. Today we recognize that electromagnetism is consistent with the theory of relativity, so the Lorentz transformations derived from electromagnetism are also consistent with relativity. Therefore, there is no need to consider the electromagnetic origin of the transformation in order to apply the transform to other aspects of special relativity. We can simply take the Lorentz transformation at face value and start from there.
Specifically, we can use the following set of Lorentz transformation equations to transform the (x, y, z, t)-coordinates of an event E observed in a rest frame S into the (x’, y’, z’, t’)-coordinates of the same event E observed in a frame S’ moving in the +x-direction with constant velocity v. Or we can use the inverse Lorentz transformation equations to transform things the other way.
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Inverse Lorentz Transformation Equations |
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x‘ = g (x – vt) |
x = g (x' + vt’) |
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y’ = y |
y = y’ |
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z’ = z |
z = z’ |
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t’ = g (t – vx/c2) |
t = g (t’ + vx’/c2) |
Gamma Factor:
Standard Configuration of Coordinates
Notice that the two coordinate systems related through these equations are
carefully oriented with respect to one another. The origins of both coordinate
systems coincide when both times begins, i.e. the event (0, 0, 0, 0) in the
system S transforms to (0, 0, 0, 0) in the system S’. The spatial
axes of the coordinate systems are parallel to one another, i.e. x is
spatially parallel to x’, y to y’, and z
to z’. The primed system S’ is moving with velocity v
in the +x-direction with respect to the unprimed system S. Conversely,
the unprimed system S is moving with velocity v in the –x’-direction
with respect to the primed system S’.
The inverse transformation equations may be derived from the original transformation equations in either of two ways. The easiest way is to recognize the physical symmetry of the situation, interchange the respective primes and unprimes, and reverse the algebraic sign in front of the velocity v. The second way is to simultaneously solve the transformation equations algebraically to obtain the inverse transformation equations.
By the way, notice that the Lorentz transformation equations reduce to the
classical Galilean
transformation equations in the limit of small velocities exactly as
required by the correspondence
principle.
Further Study:
The Lorentz transformation equations and their applications
are discussed in greater detail on pages 25-28 in Serway and on pages
20-24 in Tipler.
Questions:
Q1. Lorentz Transformations
Which of the following statements are true? (A) The Lorentz
transformation transforms from the moving frame to the rest frame. (B) The Lorentz transformation transforms the world-view of
one observer into that of another. (C) The origins of
the two coordinate systems S and S’ coincide with one another.(D) More than one of these. (E) None
of these.
Problems:
P1. Inverse Lorentz Transformation
Equations
Derive the inverse Lorentz transformation equations by solving the Lorentz
transformation equations algebraically for the respective unprimed coordinates.
P2. Gamma
(a) Invert the equation for gamma to show that 
. (b) At what velocity does gamma
become equal to 2? (b) How fast must an object travel in order for gamma to
deviate from unity by 1%? (c) At what speed is gamma equal to 1000?
P3. Star Travel
A spaceship blasts off from the earth and travels to a star in 10 years
traveling with a constant velocity of 80% the speed of light. (a) What is the
distance to the star? (b) What is the gamma factor for the spaceship? (c) What
are the coordinates of the blast-off event in both the earth frame and the
spaceship frame? (d) What are the coordinates in both frames of the arrival
event when the spaceship gets to the star? (e) What are the coordinates in both
frames of the event on the star that is simultaneous (earth time) with the
blast-off event? (f) What are the coordinates in both frames of the event on
the star that is simultaneous (spaceship time) with the blast-off event?
P4. Two Novae
Two stars explode in interstellar space. The first was seen from the earth in
1900 and was found to be 1000 light years away. The second was seen from the
earth in the year 2000 and was found to be 1500 light years away. The angle
between the two novae as viewed from the earth was 30o. (a) What is
the distance between the two events in the earth frame? (b) What is the time
interval between the two events in the earth frame? (c) If an alien spaceship
had zoomed past the earth in the year 1900 traveling toward the first nova at
half the speed of light, what would be the distance between the two novae in
the spaceship frame? (d) What would be the time interval between the two events
in the spaceship frame? (e) How much would the aliens have aged before seeing
the light from the second nova?
P5. Equations of Motion
At time t = 0 a cosmic ray passes the earth moving in the +x-direction
at a speed v = 2.9x108 m/s. At the same time, a spaceship
accelerates leaves the earth accelerating in the +x-direction with an
acceleration of a = g = 9.8 m/s2. (a) What is the equation of
motion for each of these objects in the earth frame? (b) How long does it take
the spaceship to overtake the cosmic ray? (c) What is the equation of motion
for each of these objects in the cosmic ray frame? (d) How much does the cosmic
ray age by the time the spaceship overtakes it?
P6. Ball of Light
At time t = 0 a star explodes sending out a shell of light in all
directions. The mathematical equation describing this shell is r = ct,
where r = (x2+y2+z2)1/2.
If one of the particles of matter ejected from the star travels outward at a
speed v in the +x-direction, what is the mathematical equation
describing the shell of light in the particle’s frame of reference.
P7. What is the value of gamma (g) for an object traveling with a speed of
2.9x108 m/s? (Express answer as #.#)
P8. An event has the coordinates
(1, 2, 3, 4) in the x, y, z, t-coordinate system. What are the coordinates of
that same event to one significant figure in a reference frame x’,
y’, z’, t’ moving in the +x-direction with a speed of 0.9 c?
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