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Relativity
Dr. Jay Maron



Time dilation

If you are stationary and observing a clock moving at speed V, the clock appears slow by a factor γ.

Velocity                          =  V
Speed of light                    =  C
Lorentz factor                    =  γ  =  (1-V2/C2)1/2
Time measured on stationary clock =  T
Time measured on moving clock     =  t  =  T / γ

Length contraction

Time dilation is equivalent to length contraction. Neither occurs independently of the other.

Suppose a spaceship travels from the Earth to Mars and back at a speed of V =.8 C. Suppose also that the speed of light is 1 and that the distance from the Earth to Mars is 1.

Spaceship speed                           =  V  =  4/5
Speed of light                            =  C  =   1
Lorentz factor                            =  γ  =  5/3
Distance from Earth to Mars (Earth frame) =  L  =   1
Distance from Earth to Mars (ship frame)  =  l  =  3/5  =  V t  =  L / γ
Travel time in the Earth frame            =  T  =  5/4  =  L/V
Travel time in the ship frame             =  t  =  3/4  =  T/γ

Time dilation:       t = T/γ
Length contraction:  l = L/γ
The Earth frame observes a slowdown of the spaceship clock, which is equivalent to the spaceship observing a shortening of the distance to Alpha Centauri.
Twin paradox

Suppose there are two twins, one that travels to Mars and back and the other that stays on the Earth. When the traveling twin returns he is younger than the stationary twin.

                                    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 frame observes the ship clock to be slow and the ship frame observes the Earth clock to be 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  =  5/3
Time depends on speed. When the ship reverses speed the Earth clock changes.

After the voyage the Earth clock and ship 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 ship clock reads 1.5.


Gravitational time dilation

Special relativity covers objects that are moving at high speed in flat space (zero gravity). General relativity covers gravity. Both special relativity and general relativity contribute a time dilation and the total time dilation is the sum of both.

Suppose there is a stationary observer on the Earth and an observer in orbit at 500 km and moving at 8 km/s. The stationary observer find that the orbiting clock is slower due to its speed (special relativity) and faster due to being at a higher gravitational potential (general relativity). The two effects add accoring to the figures below.


Equivalence principle

General relativity is built from the equivalence principle:

An oberver in an accelerating rocket and zero gravity experiences the same laws of physics as a stationary observer in Earth gravity.

The gravitational redshift of photons can be derived using the equivalence principle.


Gravitational redshift

If a photon travels upward by a height H it loses energy to gravity.

h  =  Planck constant
   =  6.62e-34 Joule seconds
F  =  Frequency of a photon emitted at height zero
f  =  Frequency of a photon received at height H
E  =  Energy of the photon when it is emitted
   =  h F
H  =  Height that the photon gains between being emitted and received
g  =  Gravitational acceleration
C  =  Speed of light
We assume that the relative change in frequency is small
F-f << F
Change in photon energy = Change in gravitational energy
h (F - f)  =  g H E/C2

F-f = g H E / (h C2)
    = g H F / C2
The fractional change in frequency (redshift) is
(F-f) / F  =  g H / C2
The redshift can equivalently be interpreted as a slowdown of time at height 0 compared to height H. The farther down you are in a gravitational potential the slower time passes. This is "gravitational time dilation". The fractional change in the clock rate is g H / C2.
dF   =  Change in frequency of the photon
     =  F-f
dF/F =  Fractional change in frequency
T    =  Time for one oscillation of a photon at height 0
t    =  Time for one oscillation of a photon at height H
dT   =  Change in period
     =  T-t
dT/T =  Fractional change in period

F  =  1/T

dT/T  = -dF/F
      = -g H / C2

Schwarzschild metric

We can use the above result to derive the gravitational time dilation at any distance from a point mass. This is the "Schwarzschild metric" and contains an event horizon.

G  =  Gravity constant
   =  6.67e-11 Newton meters2/kg2
M  =  Mass of the point mass
R  =  Distance from the point mass
g  =  Gravitational acceleration
   =  - G M / R2
T  =  Time dilation factor as a function of R

dT/T  =  - g H / C2
      =  (G M / R2) * dR / C2

dT / T = (G M / C2) dR / R2
ln(T)  =  (G M / C2) / R


Gravitational time dilation

Objects in a gravitational well experience time more slowly than objects in flat space.

M  =  Mass of gravitating object
G  =  Gravity constant
R  =  Distance from the gravitating object
g  =  Gravitational acceleration
   =  G M / R2
C  =  Speed of light
Rs =  Schwarzschild radius for the gravitating object
   =  Radius of the event horizon
   =  2 G M / C2
T  =  Time for one oscillation of a photon as a function of R
t  =  Time for one oscillation of a photon at R = Infinity
dR =  Change in distance from the gravitating object
   =  H                (The height from the above calculation)

The time dilation factor is
T  =  t (1 - 2 G M / (R C2))  =  t (1 - Rs/R)-1/2
If the photon is far from the gravitating object and if the change is R is small then this formula reduces to the approximation given above.
T     =  t (1 - 2 G M / (R C2))-1/2
      ~  t (1 + G M / (R C2))
dT    ~  G M t dR / (R2 C2)
dT/T  =  G M dR / (R2 C2)             For small redshifts we approximate T~t so that dT/T ~ dT/t.
      =  g H dR / C2
      =  g H / C2
If you are far from a black hole but not at infinity then the time delay factor is
d  =  1 - (1-Rs/R)½
   ~  .5 Rs / R

Timeline
1859  Mercury's orbit is found to deviate from Newtonian gravity
1905  Theory of special relativity developed
1915  Theory of general relativity developed
1915  Schwarzschild solves the metric for a point mass and find that
      general relativity implies an event horizon (black hole)
1919  Eddington measures the deflection of starlight during a solar eclipse
      and confirms the prediction of general relativity
1922  Friedmann finds a solution in which the universe may expand or contract.
      Einstein adds a cosmological constant to general relativity.
1927  Lemaitre showed that static solutions of the Einstein equations,
      which are possible in the presence of the cosmological constant,
      are unstable, and therefore the static universe envisioned by Einstein
      could not exist (it must either expand or contract)
1929  Hubble finds that the universe is expanding
1963  Kerr solves the metric for a spinning black hole
1966  Shapiro measures the gravitational time delay of photons
1974  Hulse and Taylor analyze a double pulsar and confirm that it emits
      gravitational waves consistent with general relativity
1958  Mossbauer effect discovered, allowing for high-precision measurements of
      photon energy
1959  Pound and Rebka use the Mossbauer effect to measure the gravitational redshift
2007  The Gravity Probe B satellite detects frame-dragging and the geodetic effect

The film "Interstellar"

In the film "Interstellar" the astronauts experience extreme gravitational time dilation. This is likely accompanied by extreme tidal forces.

If we assume that the source of warping of space is a black hole then

G  =  Gravitational constant
M  =  Mass of the black hole
C  =  Speed of light
Rs =  Radius of the event horizon of a black hole (Schwarzschild radius)
   =  2 G M / C2
D  =  Gravitational time dilation factor
   =  (1 - 2 G M / (R C2))
   =  1       for R -> Infinity
   =  0       at the black hole event horizon
A  =  Gravitational acceleration
   =  - G M / R2
Z  =  Gradient of the acceleration
   =  dA/dR
   =  - 2 G M / R3
   =  - Rs C^2 / R3
Suppose we consider D=.5 to be onset of extreme time dilation. D=.5 corresponds to R = (4/3) Rs. In other words you need to be quite close to the black hole. The time dilation experienced in the film was even more extreme than this.

Suppose we consider Z=1 to be the largest tidal force that can be endured over a sustained period of time. If

Z=1      at     R = (4/3) Rs
then the mass of the black hole is
Z  = (27/256) C6 / (G M)2

M  =  (27/256)½ C3 / G / Z½
   ~  .3 C3 / G / Z½
   ~  1.2e35 kg
   ~  60000 solar masses
To experience extreme time dilation while not getting spaghettified you need a large black hole.
Deflection of light by gravity

Eddington's original photographic plate

General relativity bends light by twice the amount predicted by Newtonian mechanics.


Gravitational lensing
Artist's conception of a black hole
A lensed galaxy

Precession of Mercury's orbit

In Newtonian gravity, if 2 gravitatating objects are ideal point masses then the orbits don't precesss, and in general relativity they do. In 1859 Mercury's orbit was found to precess, and the precession was found to be in accord with general relativity.


Photons near a black hole

A photon launched tangental to a black hole will orbit as a circle if it is at 3/2 times the event horizon radius. If it is closer it will spiral in and if it is further it will escape.

The circular orbit is unstable. If the photon deviates from the circle it will either fall into the hole or escape.


Spinning black hole

A spinning black hole drags the space around it. Photons orbiting the hole in the direction of the spin appear to move faster than photons orbiting counter to the spin.


Spaghettification

Objects near a black hole experience extreme tidal forces.


Gravitational waves

A pulsar and a white dwarf inspiral and merge
Gravity waves generated by 2 pulsars

Two orbiting objects emit gravitational waves. If the objects are large enough the will lose energy, inspiral, and merge.


Detecting gravitational waves

Passing gravity waves acting on a ring
Laser interferometer
Detecting gravity waves in space
Inspiral of the Hulse-Taylor pulsar.

Sensitivity of detectors


Stellar lifetime

Hubble image of Sirius A and B
Artist's conception of Sirius A and B
Artist's conception of the sun and Pegasi A and B

Sirius B and Pegasi B are white dwarfs and they appear as the small blue dots in the images.

Stellar lifetime
Stellar lifetime

Evolution of the sun from its birth to a red giant
Artist's conception of the Earth and sun 5 billion years from now


Life cycle of stars

M  =  Star mass in solar masses

   Case        Death           Remnant

  M <  9       Red giant       White dwarf
9 < M < 20     Supernova       Neutron star
  M > 20       Supernova       Black hole

Neutron star

After a star runs out of fusion fuel the core contracts. Electron pressure can hold up the core unless the mass exceeds 1.4 solar masses (the Chandrasekhar limit), at which point electrons combine with protons to produce neutrons and the core collapses into a neutron star. The energy released powers a supernova and ejects the surface layer.

Neutron pressure can hold up a neutron star unless the mass exceeds 3 solar masses, at which point it collapses into a black hole.

Object              Mass       Radius    Core      Core
                  (Sun=1)      (km)     density    state
                                        (g/cm3)
Earth               .000003     6371       12      Iron
Jupiter             .00096     69900       25      Iron
Red dwarf           .08        70000      100      Plasma
Sun                1          696000      150      Plasma
White dwarf max    1.4          3500      1e8      Electron matter
Neutron star max   3              13     8e14      Neutron matter
Black hole min     3               9     2e15      The radius refers to the event horizon radius


               Density (g/cm3)
Water                 1.0
Iron                  7.9
Osmium               22.6    Densest element
Earth core           12
White dwarf max     1e8
Nuclear matter     2e14
Neutron star max   8e14      Maximum core density before collapsing into a black hole
Black hole         2e15      Density of a 3 solar mass black hole (mean density inside the event horizon)

Neutron star

Neutron star
White dwarf radius as a function of mass


Magnetar

All neutron stars have extreme magnetic fields, and the ones with the most extreme fields are called magnetars. Magnetars have a field of between 108 and 1011 Tesla. A neodymium magnet has a field of 1.25 tesla and a magnetic energy density of 4.0e5 J/m3. A magnetar's 1010 tesla field, by contrast, has an energy density of 4.0e25 J/m3. The magnetic field of a magnetar is lethal at a distance of 1000 km due to the strong magnetic field distorting the electron clouds of the subject's constituent atoms, rendering the chemistry of life impossible. At a distance halfway to the moon, a magnetar could strip information from the magnetic stripes of all credit cards on Earth. As of 2010, they are the most magnetic objects ever detected in the universe.

As described in the February 2003 Scientific American cover story, remarkable things happen within a magnetic field of magnetar strength. "X-ray photons readily split in two or merge together. The vacuum itself is polarized, becoming strongly birefringent, like a calcite crystal. Atoms are deformed into long cylinders thinner than the quantum-relativistic de Broglie wavelength of an electron." In a field of about 105 teslas atomic orbitals deform into rod shapes. At 1010 teslas a hydrogen atom becomes a spindle 200 times narrower than its normal diameter.

                                Magnetic field in Teslas

Field generated by brain             10-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Large Hadron Collider magnets       8.3
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014

Compact objects
                     Mass     Radius  Schwarzschild  Surface grav   Surface tide     Escape    Core    Core temp
                    (Sun=1)    (km)   radius (km)      (m/s2)         (m/s2/m)       speed    density  (MKelvin)
                                                                                     (C=1)    (g/cm3)
Sun                      1   696000         3.0           279.4           .00000039    .0021    150       15
Red dwarf            .08      70000          .24         2170             .00000031    .0018    100       13
Jupiter              .00096   69900          .0028         24.8           .00000037    .00020    25         .036
Earth                .000003   6371          .000009        9.8           .0000015     .000037   12         .006
White dwarf max          1.4   3500         4        15000000            4.3           .034      e8       10
Neutron star max         3       13         9          2.4e12    180000000             .82     8e14        1
Black hole min           3        9         9          4.9e13    550000000            1        2e15        -
Milky Way BH       4200000        -  12400000         3600000            .00029       1        1050        -
Andromeda BH     200000000        -     5.9e8           76000            .00000013    1         .46        -
APM-08279 BH   23000000000        -    6.8e10             660             9.7e-12     1         .00003     -


Gravity constant          =  6.67e-11 N m2 / kg2
Sun mass                  =  1.989e30 kg
Schwarzschild radius      =  2 G M / C2
Sun Schwarzschild radius  =  2.95 km
Iron nucleus density      =  2.3e14 g/cm3
Neutron star core pressure=  1.6e35 Pascals
The tidal magnitude is the change in gravitational acceleration per distance from the gravitating object.
Tidal magnitude   Effect on a human
   (m/s2/m)
      .1          Minimum perceptible
     1            Tangibly perceptible
    10            Uncomfortable
    50            Threshold for blackout, where the heart cannot pump blood to the brain
If you are standing on the surface of a white dwarf then the tidal magnitude is tangible but not uncomfortable. The gravity, however, is enough to squash you.

The larger the mass of a black hole, the smaller the tidal magnitude at the Schwarzschild radius. The tidal magnitude is imperceptible at the surface of the Milky Way supermassive black hole.

The largest known black hole is AMP-08279.

For the above calculations we have assumed Newtonian physics, which is okay if the escape speed is substantially less than the speed of light. This applies for stars, planets, and white dwarfs, but not for neutron stars and black holes. For these objects you need general relativity and the Newtonian numbers should be regarded as approximate.


Jet

Matter falling into a black hole forms an accretion disk. The disk generates a magnetic field and the field ejects material from the disk in a jet. Most of the material falls into the hole and some is ejected in the jet.


Lorentz transform

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 = T
This 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-V2/C2)

x = Z (X - VT)
t = Z (T - VX/C2)
This is the "Lorentz transform".

The time transform depends both on time and space. Time is not absolute.

If V << C,

Z  =  (1-V2/C2)
   ~  1 + .5 V2/C2
If you are on an airplane moving at the speed of sound, V = 300 m/s and
Z  ~  1 + 5e-13
The effects of realativity are hard to notice at this speed.
Speed of light

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 T
In Galilean relativity, the coordinate of the photon in the moving frame is
t  =  T
x  =  X - V T  = CT - VT  =  (C-V) t
In 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/C2)   =  Z (T  - VT/C)  =  Z T (1-V/C)
The speed of the photon in the moving frame is
c = x/t = C
The photon moves at speed C in the moving frame. The Lorentz transform preserves the speed of light.
Lorentz transform and its inverse

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 / C2)             T = Z (t + V x / C2)

Velocity addition

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/C2)       Velocity addition formula
In Galilean relativity,
U  =  V + v
The 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/C2)   =  Z (t + Vvt/C2)

U  =  X/T  = (V + v) / (1 + Vv/C2)

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.


Invariant interval

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 - C2 (T2-T1)2
Shift 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  =  X2 - C2 T2
s  =  x2 - C2 t2
The Lorentz transform preserves the invariant interval.
S2  =  s2
If a photon is created in the first flash and arrives at the instant of the second flash,
S2  =  s2  =  0
The 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.
Causality

Suppose that it's possible to travel from Flash #1 to Flash #2 at speed V, where V < C. Then

S2 > 0    and    s2 > 0
If S2 > 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

S2 < 0    and    s2 < 0
We 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.


Proper time and time dilation

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 / C2)
   =  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).
Time dilation

Visualization

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-V2/C2)
D can be evaluated by constructing a right triangle.
D2   =  d2 + L2
     =  d2 + V2 D2 / C2

D2 (1-V2/C2) = d2

D = Q d

T = Q t
More time passes in the stationary frame than in the ship frame by a factor of Q.
Timeline
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

Relativistic velocity addition
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/C2)
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 = C
The 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?


Magnetic force

The magnetic force is inconsistent with Galilean relativity because it depends on the speed of the charge. Special relativity resolves the inconsistency.

Suppose two equal charges are moving in parallel. In CGS units,

Electric charge                     =  Q
Distance between the charges        =  R
Velocity of the charges             =  V
Speed of light                      =  C
Mass of the charges                 =  M
Lorentz gamma factor                =  Z    =  (1-V2/C2)
Electric force in the moving frame  =  Fe   =  Q2 R-2
Electric accel. in the moving frame =  Amove=  F/M  =  Q2 R-2 M-1
Electric accel. in the rest frame   =  Arest=  Amove Z-2  =  Q2 R-2 (1-V/C2) M-1
Electric force in the rest rame     =  Q2 R-2 - Q2 (V/C)2 R-2
                                    =  Fe + Fm
Magnetic force between the charges  =  Fm  = -Q2 (V/C)2 R-2

Two stationary charges experience an electric repulsion. If the charges move in parallel, time dilation slows down the electric force, which can be interpreted as an added attractive force. This attractive force is the magnetic force.


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