In a stationary frame, let an event have space and time coordinates
S = (X,T)In a frame moving at speed +V, the event has coordinates
s = (x,t)In classical physics, s is related to S by
x = X - V T t = TThis is "Galilean relativity". In this transform, t doesn't depend on X. Time is absolute.
In special relativity,
C = Speed of light in the stationary frame c = Speed of light in the moving frame Z = 1 / Squareroot(1-V^2/C^2) x = Z (X - VT) t = Z (T - VX/C^2)This is the "Lorentz transform".
The time transform depends both on time and space. Time is not absolute.
If V << C, Z = (1-V^2/C^2)^(-1/2) ~ 1 + .5 V^2/C^2If you are on an airplane moving at the speed of sound, V = 300 m/s and
Z ~ 1 + 5e-13The effects of realativity are hard to notice at this speed.
Suppose a photon is created in the stationary frame at S=(0,0) and travels to the right at speed C. Its coordinates in the stationary frame are
X = C TIn Galilean relativity, the coordinate of the photon in the moving frame is
t = T x = X - V T = CT - VT = (C-V) tIn the moving frame the photon moves to the right at speed (C-V).
In special relativity, the coordinate of the photon in the moving frame is
x = Z (X - VT) = Z (CT - VT ) = Z T (1-V/C) C t = Z (T - VX/C^2) = Z (T - VT/C) = Z T (1-V/C)The speed of the photon in the moving frame is
c = x/t = CThe photon moves at speed C in the moving frame. The Lorentz transform preserves the speed of light.
The Lorentz transform gives the coordinates in the moving frame as a function of the coordinates in the stationary frame. In other words,
(x,t) as a function of (X,T).We can invert the transform to express the coordinates in the stationary frame as a function of the coordinates in the moving frame. In other words,
(X,T) as a function of (x,t)The inverse transform can be obtained by replacing +V with -V in the forward transform.
Lorentz transform: Inverse transform: x = Z (X - V T) X = Z (x + V t) t = Z (T - V X / C^2) T = Z (t + V x / C^2)
Suppose a moving spaceship fires a cannon. What is the speed of the cannonball in the stationary frame?
V = Spaceship velocity with respect to the stationary frame
v = Cannonball velocity with respect to the spaceship
U = Velocity of the cannonball in the stationary frame
(V + v) / (1 + Vv/C^2) Velocity addition formula
In Galilean relativity,
U = V + vThe cannonball is fired at (X,T) = (x,t) = (0,0). The coordinates of the cannonball in the spaceship frame are x = vt. In the stationary frame, the projectile is at
X = Z (x + Vt) = Z (vt + Vt) T = Z (t + Vx/C^2) = Z (t + Vvt/C^2) U = X/T = (V + v) / (1 + Vv/C^2) If v < C, then U < C If v = C, then U = C (if the cannonball is replaced by a photon)If an object moves at less than the speed of light in one frame, then it moves less than the speed of light in all frames.
If an object moves at the speed of light in one frame, it moves at the speed of light in all frames.
Suppose you see two simultaneous flashes of light.
(Space,time)
Flash #1 = S1 = (X1,T1)
Flash #2 = S2 = (X2,T2)
The "invariant interval" between S1 and S2 is
S = (X2-X1)^2 - C^2 (T2-T1)^2Shift the coordinate system so that X1=T1=0
(Space,time)
Flash #1 = S1 = (0,0)
Flash #2 = S2 = (X,T)
In the moving frame,
(Space,time)
Flash #1 = s1 = (0,0)
Flash #2 = s2 = (x,t)
S = X^2 - C^2 T^2
s = x^2 - C^2 t^2
The Lorentz transform preserves the invariant interval.
S^2 = s^2If a photon is created in the first flash and arrives at the instant of the second flash,
S^2 = s^2 = 0The path of a photon from one place to another always has an invariant interval equal to 0. This is equivalent to saying that the speed of light is constant in all frames.
Suppose that it's possible to travel from Flash #1 to Flash #2 at speed V, where V < C. Then
S^2 > 0 and s^2 > 0If S^2 > 0, we say that the two flashes are "causally connected" and that the interval is a "timelike interval". It is possible for Flash #1 to have an effect on Flash #2.
If two flashes are causally connected in one frame then they are causally connected in all frames.
Suppose it's not possible to travel from Flash #1 to Flash #2, even if you are moving at the speed of light. Then
S^2 < 0 and s^2 < 0We say that the two flashes are "causally disconnected" and that the interval is a "spacelike interval". It is not possible for Flash #1 to have an effect on Flash #2.
If two flashes are causally disconnected in one frame then they causally disconnected in all frames.
The Lorentz transform preserves causality.
If two flashes are causally connected then there exists a frame where they occur at the same point in space. The time between the flashes in this frame is the "Proper time". The observed time between flashes is longer in all other frames (time dilation).
Let the coordinates of two flashes in the stationary frame be
Flash #1 = S1 = (0,0) Flash #2 = S2 = (0,T)In a moving frame,
Flash #1 = s1 = (0,0) Flash #2 = s2 = (x,t)The proper time is T. We use the forward Lorentz transform to obtain the position and time for flash #2 as observed in the moving frame.
t = Z (T - V 0 / C^2) = Z T t > T (time dilation)The time measured between two events is smallest in the frame where the events occur at the same place (the proper time).
V = Velocity of the ships
C = Speed of light
d = Distance between the ships, which is the same in both the ship and
stationary frames
= Distance the light beam travels as it goes from the top ship to the bottom ship,
in the ship frame
D = Distance the light beam travels as it goes from the top ship to the bottom ship,
in the stationary frame
t = Time required for the light signal to travel from the top ship to the bottom ship
in the ship frame
= d / C
T = Time required for the light signal to travel from the top ship to the bottom ship
in the stationary frame
= D / C
L = Distance the ships move as the light signal travels from the top ship to the
bottom ship, in the stationary frame.
= V T
= V D / C
Q = Lorentz time dilation factor
= (1-V^2/C^2)^(-1/2)
D can be evaluated by constructing a right triangle.
D^2 = d^2 + L^2
= d^2 + V^2 D^2 / C^2
D^2 (1-V^2/C^2) = d^2
D = Q d
T = Q t
More time passes in the stationary frame than in the ship frame by a factor of Q.
Time dilation is equivalent to length contraction. Neither occurs independently of the other. Suppose a spaceship travels to Alpha Centauri.
L = Distance from Earth to Alpha Centauri in the Earth frame = 4.4 light years l = Distance from Earth to Alpha Centauri in the spaceship frame C = Speed of light V = Speed of a spaceship on the way from the Earth to Alpha Centauri = 4/5 C Z = Lorentz factor = (1-V^2/C^2)^(-1/2) = 5/3 T = Time for the journey in the Earth frame = L / V = 5.5 years U = Time for the journey in the spaceship frame (time dilation) = T / Z = 3.3 yearsIn the spaceship frame, the distance traveled is
l = V t = L / Z = 2.6 light yearsThe Earth frame observes a slowdown of the spaceship clock, which is equivalent to the spaceship frame observing a shortening of the distance to Alpha Centauri.
Suppose a spaceship travels from the Earth to Mars and back at a speed of V=4/5. Suppose also that the speed of light is 1 and that the distance from the Earth to Mars is 1.
V = Spaceship speed = 4/5 C = Speed of light = 1 Q = Lorentz factor = (1 - V^2/C^2)^(-1/2) = 5/3 D = Distance from Earth to Mars in the Earth frame = 1 T = Time for the journey from Earth to Mars in the Earth frame = D/V = 5/4 t = Travel time from Earth to Mars in the frame of the moving ship (Time dilation) = T/Q = 3/4 d = Distance from Earth to Mars in the frame of the moving ship (Length contraction) = V t = 3/5From the point of view of the Earth, the trip appears shorter for the spaceship because time passes more slowly on the ship.
From the point of view of the ship, the trip appears shorter because the distance between Earth and Mars is Lorentz contracted.
Earth Earth Ship Ship
Timeline of Ship Ship frame, frame, frame, frame,
spaceship velocity position Earth Ship Earth Ship
clock clock clock clock
Before launch 0 0 .0 .0 .0 .0
Departs Earth +.8 0 .0 .0 .0 .0
Arrives at Mars +.8 1 1.25 .75 .45 .75
Stops at Mars 0 1 1.25 .75 1.25 .75
Departs Mars -.8 1 1.25 .75 2.05 .75
Arrives at Earth -.8 0 2.5 1.5 2.5 1.5
Stops at Earth 0 0 2.5 1.5 2.5 1.5
When the ship is moving, the Earth observer thinks the ship clock is slow
and the ship observer thinks the Earth clock is slow. Upon arrival at Mars,
the Earth observer sees his clock at T=1.25 and the ship clock as t=.75.
The ship observer sees his clock as t=.75 and the Earth clock as T=.45.
The time dilation factor for both observers is the same.
1.25 / .75 = 5/3 .75 / .45 = 5/3Time depends on speed. When the spaceship reverses speed the Earth clock changes.
After the voyage the Earth clock and spaceship clock are at the same point in space and they are moving at the same speed (V=0). At this point the Earth clock reads 2.5 and the spaceship clock reads 1.5.
1769 Robison measures the inverse square law for electric charges
1800 Volta invents the battery, enabling the generation of large electric currents
1803 Young discovers the diffraction of light, suggesting that light is a wave
1820 Orsted finds that an electric current produces a magnetic field
1826 Ampere finds that electric currents attract each other
1831 Faraday finds that a changing magnetic field produces an electric field
1861 Maxwell finds that a changing electric field produces a magnetic field
1861 Maxwell develops the "Maxwell's equations", unifying electricity and magnetism
1864 Maxwell finds that light is an electromagnetic wave
This theory contained a paradox, that the speed of light is invariant
1884 Heaviside invents the vector calculus and uses it to simplify Maxwell's equations
1887 Hertz achieves the first detection of electromagnetic waves
1887 Michelson-Morley experiment finds that the speed of light is invariant
1889 Heaviside publishes the force law for a charge moving in a magnetic field
1892 Lorentz discovers the "Lorentz transform" for special relativity
This offered an explanation for the Michelson-Morley experiment
1904 Lorentz finds that the "Lorentz transform" resolves the paradoxes of
Maxwell's equations
1905 Einstein and Poincare each publish a complete formulation of the theory
of special relativity
1915 Einstein develops the theory of general relativity
V = Velocity of spaceship v = Velocity of cannonball fired from the ship, in the ship frame U = Velocity of the cannonball in the rest frame C = Speed of light U = (V + v) / (1 - Vv/C^2)This equation forbids anything from reaching the speed of light.
If V < C and v < C then U < C.If the cannonball is a photon,
v = C U = CThe speed of light is the same in both the ship and the stationary frame
Suppose a multistage rocket is such that: The first stage accelerates from rest to .5 C. The second stage accelerates from rest to .5 C, in the frame of the first stage. Etc.
If the rocket has 5 stages, what is the velocity of each stage in the rest frame? What is the Lorentz factor of the final stage?