A photon with wavelengthcan be considered as a particle with mass zero and momentumwhere h = Planck's constant. When the photon collides with a stationary electron, it is scattered elastically and its wavelength increases fromto

Conservation of momentum gives

(1)

(2)

Multiplying (1) and (2) bythroughout and usinggives(3) and(4)

Rewriting (3) as(5), squaring and adding (4) and (5) gives

(6)

Conservation of energy gives

Squaring both sides of the last equation gives(7)

(7)-(6) and simplifying the result gives

Dividing byand rearranging gives

Butandso

]]>Suppose the two inertial frames O and O' coincide at t=t'=0. At this instant a light pulse is emitted and produces an event E at coordinatesin the inertial frame O andin the inertial frame O'.

andare related bySince O' is moving along the x axis of O,andhence

Since light travels in straight lines, the relationship betweenandwill be linear so we can write(1) and(2)

Sincewhen light ray is emitted, from (1)

(1) becomes(3)

Substitute (2) and (3) these expressions intoto get

Equating coefficients of (4)

Equating coefficients of (5)

Equating coefficients of (6)

Square (5) to give (7)

From (4)and from (6)

Substitute these into (7) to obtain

Expanding and simplifying gives

Then(Take the positive square root so thatandflow in the same direction).

From (5)

Thenand

The transformation is symmetrical, soand

]]>Two observers observing two events to have a difference in coordinateandwil obtain the same values forand

this difference is labelledand is called the invariant spacetime interval and can be positive, negative or zero. The distinction is important.

Ifthen the event occurring earliest may not be a possible cause of the later event. The distance between the two events is greater than can be covered by a light particle in a time

Ifthen the event occurring earliest may be a possible cause of the later event. The distance between the two events is equal to the distance that can be covered by a light particle in a time

Ifthen the event occurring earliest may be a possible cause of the later event. The distance between the two events is less than can be covered by a light particle in a time

These different possibilities can be illustrated on a spacetime diagram.

The event at the cone vertex may be the cause of any event in region (1) in the diagram below, for whichbut not in region (2) for which

In region (2) different observers may not even agree about the order of events.

Similarly the event at the origin lies in the regionfor event A below, so might have been caused by A, but not by event B, since it lies in the regionfor event B.

]]>We can write these expressions as generalized dot products.

The transformation fromtois

The determinant of the transformation matrixby expanding along the top row is

The dot product defined above preserves distances under the Lorentz transformation. Any vector satisfying the distance preserving property is called a 4 – vector. Examples are momentum and energyand electric/magnetic fields or current and charge density.

]]>Length contraction is another of the strange consequences of special relativity. Moving bodies appear to be shorter than stationary bodies. Of course, if the moving body is a spaceship, it will not appear shorter to the astronauts on board the spaceship, just as time on board the spaceship will not appear to pass more slowly.

If two events 1 and 2 occur at the points in spacetime with coordinatesand and these are the measuring of the ends of a ruler which is stationary in O', and we assume that these events happen at the same time in O so thatthen the distance between the two events isin O'. We can use the Lorentz transformation to find the distance between the two events as measured in O.

(1)

(2)

(2)-(1) gives

Hence we can writewhereandare the lengths of the ruler in O and O' respectively.

]]>In Newtonian mechanicsandis constant so

In special relativity, mass plays more of a resistance to acceleration role, and has a different value in the direction of acceleration than at right angles to it.

In special relativity the momentum of a particle of rest massiswhere

If the acceleration is perpendicular tothen and

If the acceleration is in the direction of

]]>The lines shown in black are possible paths of light rays and the curves in red are possible paths of matter particles. Any path passing from below the x axis to above the x axis goes from past (t<0) to future (t>0). A path which passes through the origin can be considered to pass through the the origin of an intertial frame at

Different inertial frames moving with constant velocity relative to each other may be represented on the same Minkowski diagram. If an inertial frame O' with axesis moving with velocity along the x axis of the inertial frame O above then the Lorentz transformation gives the coordinates in O' of any events with coordinatesin O. They are

On the x' axis,and on theaxisso the equations of the x' axis in O isand the equation of theaxis in O is

]]>The observer on the right is stationary in his reference frame, and the observer on the left sees the other moving to the right with speed

The observer on the right bounces a light signal between the ceiling and the floor. It takes a timeto return to the floor, so that

The observer on the left sees the moving framer of reference moving to the right, and will see the ray of light travel a distancein a timewhereHe will observe the ray of light to take a timeto return to the floor, so that

Substituteandinto

Square both sides to give

hence

Move the last term to the left.

Divide byto give

Factorise withto give

Hence

]]>The clock carried by observer O ticks at timeseach tick results in a light signal being sent to O' which is received at timesand instantly reflected back to O.

Because the world line of O' in the inertial frame O is a straight line, there will be a linear relationship betweenandand because it passes through the origin, the relation is of the form so thatSimilarly, when light signal is received in O' and reflected,

The signals are received atso that

The signals take equal times to travel from O to O' and back, so the first signal is received by O' at a time in Os frame given and

Hence the speed of O' relative to O is

so

Henceand the frequencyof light received by a moving observer is reduced by the factor

]]>The components of velocity of the light pulse in O are

(1)

(2)

Dividing (2) by (1) gives

For the special case when the pulse of light is emitted along the y' axis of O' so that and

The effect of the aberration is that as the speed of O' increases,the direction of the light pulse in O becomes more aligned with the x axis

Ifthen forrespectively then the angles %theta that the light pulse makes with the x axis in O are 84, 53 and 26 degrees respectively.

]]>