14. Special theory of relativitySavchenko Formulas, chapter 14 of 14, 16 formulas

$u'$

velocity of the body in the moving frame

v

velocity of the moving frame

u

velocity of the body in the rest frame

Velocities along one line add so that the result never exceeds $c$, and the speed of light stays $c$ in every frame. For $u', v \ll c$ the formula turns into Galilean addition. The transverse component changes too, because of time dilation.

$\nu$

frequency in the source frame

$\nu'$

frequency at the observer

$\beta$

closing speed as a fraction of $c$

$\alpha$

angle between the velocity and the ray

Unlike sound, only the relative velocity matters for light. Time dilation of the source joins the classical factor $1 + \beta\cos\alpha$, so even transverse motion shifts the frequency by $\gamma$. Reflection from a moving mirror gives the factor twice.

$\theta$

angle of the ray to the velocity in one frame

$\theta'$

the same angle in a frame moving at $\beta c$ towards it

$\beta$

$v/c$

This is velocity addition for light, $u_x = c\cos\theta$. For a moving observer the rays crowd towards the direction of motion, and the radiation of a fast particle is gathered into a narrow cone of angle about $1/\gamma$.

v

speed

c

speed of light

The factor by which time intervals stretch and lengths along the motion shrink for a moving body. For $v \ll c$ it is $1 + v^2/2c^2$, as $v \to c$ it grows without bound.

l₀

proper length, in the rest frame of the body

l

length of the moving body along its velocity

Only dimensions along the velocity shrink, transverse ones do not change. So the area of a plate moving in its own plane falls $\gamma$ times and the surface charge density on it grows by as much.

$\Delta\tau$

proper time, on a clock moving with the body

$\Delta t$

the same interval on the rest clocks

Moving clocks run slow. So an unstable particle with proper lifetime $\tau$ travels $\gamma v\tau$ in the laboratory, not $v\tau$. Proper time is shorter than any other between the same two events.

x, t

position and time of an event in the rest frame

$x', t'$

the same in the frame moving at $v$ along $x$

Time dilation, length contraction and velocity addition all follow from them. Events simultaneous in one frame and $\Delta x'$ apart are separated by $\gamma v\Delta x'/c^2$ in another, the relativity of simultaneity. The interval $c^2t^2 - x^2$ is the same in every frame.

$\parallel, \perp$

components along the frame velocity and across it

v

velocity of the moving frame

The electric and magnetic fields are one field seen from different frames. A frame moving through a pure magnetic field sees an electric field $\gamma vB$, and the other way round. $E^2 - c^2B^2$ and $\mathbf{E}\cdot\mathbf{B}$ do not change, so for instance a pure electric field cannot be turned into a pure magnetic one. In Gaussian units $v$ becomes $v/c$ and $v/c^2$ becomes $v/c$.

v

velocity of the charge

E

its electric field

Q

charge

The field of a uniformly moving charge is the Coulomb field of its rest frame, transformed. Across the velocity it is $\gamma$ times stronger, along it $\gamma^2$ times weaker, and the magnetic field at any speed is $\mathbf{v}\times\mathbf{E}/c^2$.

$\rho$

charge density

j

current density along the velocity

A neutral wire carrying a current is charged in a frame moving along it, because the spacings of the electrons and of the ions contract differently. That is how a magnetic force on a moving charge in one frame becomes an electric one in another.

m

mass of the particle

v

its velocity

The momentum grows without bound as $v \to c$ while the speed stays below $c$. From $p$ the speed is $v = pc^2/E$ or $v = c/\sqrt{1 + (mc/p)^2}$.

p

relativistic momentum

F

force, for example $eE$ in an electric field

A force changes the momentum, not the mass times the acceleration. In a uniform electric field the momentum grows linearly with time and the speed approaches $c$ without reaching it. The transverse momentum in a longitudinal field is conserved, so the transverse speed falls as the energy grows.

R

radius of the circle

q

charge of the particle

B

magnetic induction

A magnetic field does not change the energy, so $\gamma$ is constant and the particle moves on a circle as in the non-relativistic case, only with momentum $\gamma mv$. The track radius and the field give the momentum at once, and the revolution frequency falls as the energy grows.

m

mass of the particle

E₀

rest energy

K

kinetic energy

The total energy includes the rest energy. Energy given off or taken up by a body changes its mass by $\Delta E/c^2$, so the mass of a system differs from the sum of its parts by the binding energy over $c^2$. For $v \ll c$ the kinetic energy is $mv^2/2$.

E

total energy

p

momentum

m

mass

A massless particle, a photon, has $E = pc$. An ultrarelativistic particle has $E \approx pc$, a slow one $E \approx mc^2 + p^2/2m$. In decay and collision problems the relation is written for every particle together with energy and momentum conservation.

M

mass of the system, its energy in the centre of mass frame over $c^2$

E_i, p_i

energies and momenta of the particles

This quantity is the same in every frame and is conserved in decays and collisions. It is computed in whichever frame is easier, for instance in the laboratory before the reaction and in the centre of mass frame after it. Hence the threshold of particle production, $Mc^2$ is at least the sum of the rest energies of the products.