Relativity, special It deals with measurements made in reference frames moving with uniform velocity (v~ c) with respect to one another. When the relative velocity between reference frames approach the speed of light, the Galilean transformations fails. Another set of transformations , called the Lorentz transformations, are applicable. The classical concepts of length, time, mass break down. These. are no longer invariant quantities, but depend upon the reference frame from where the object is viewed. The theory emerged from the pioneering experiment carried out by Michelson and Morley and theoretical work by Lorentz, Poincare and others, were fully developed by Albert Einstein.

Around the end of nineteenth century, physicists observed that under Galilean transformations the Maxwell’s equations are not covariant, that is, they do not retain their form. Lorentz discovered a set of transformations that keep the form of Maxwell’s equations. Michelson and Morley carried out a series of experiments to detect ether, supposedly an all pervading medium in which light waves propagate. The negative result of this experiment proved that, (i) the medium ether does not exist, and (ii) the velocity of light does not depend upon the motion of the source or observer

Einstein in the year l905, wrote a paper in which he comprehensively stated two postulates of relativity, and further deduced the full consequences of Lorentz transformations. The postulates of relativity are the following.

I. The principle of relativity: All physical laws have the same form in all inertial (non accelerating) frames.

II The principle of constancy of speed of light: The speed of light in free space is the same in all inertial frames. It does not depend upon the motion of the source or the observer.

Lorentz transformations: Consider two frames O at rest and O¢ moving with velocity v with respect to O along the common x/x¢ axis (see fig. r1). The y axis of O is parallel to y¢ axis, and the z axis of O is parallel to z¢ axis of O¢ . An event in O with space time coordinates (x,y,z,t) will be seen in O¢ with coordinates (x¢ ,y¢ ,z¢ ,t¢ ). The transformation equations for coordinates in O frame to that in O¢ are,

x¢ = g (x-vt)

y¢ = y

z¢ = z

t¢ = g {t-(vx/c²)} (r7)

Where g = (1-b ²)^-1/2 with b = v/c

is called the Lorentz factor. The reverse transformations that is from coordinates of O¢ to O frame are obtained by replacing undashed by dashed quantities and vice versa, and replacing +v with -v.

Consequences of Lorentz transformations; Lorentz Fitzgerald contraction: Let L₀ is the length of a rod in a frame in which it is at rest (called the proper length). If rod is viewed from another frame which is moving with constant velocity v along the length of the rod, the length of the rod will appear shorter, by a factor g .

L= L₀/g (r8)

where L is the length of the rod from the moving frame.

Time dilation: If two events have occurred at the same point with time interval D t , in the rest frame, the same will be seen with time interval D t from a moving frame, such that,

D t = g D t (r9)

As g is greater than 1, the time interval will appear larger. D t is called the proper time. Moving clocks appear to run slow , as a consequence of time dilation.

Relativistic velocity transformations: The components of velocity of a body in O frame, u_x, u_y, u_z, are transformed in the O¢ as,

u_x ¢ = (u_x - v) [1- (v u_x/ c² )]^-1

u_y ¢ = u_y g ^-1 [1- (v u_x/ c² )]^-1

u_z ¢ = u_z g ^-1 [1- (v u_x/ c² )]^-1 (r10)

Compare these transformations with that obtained in Galilean transformations.

Increase in mass with velocity: Let the mass of a body in a frame where the body is at rest be m₀. From another frame, in which the body is moving with velocity v, it mass appears to have increased to m,

m = m_0g (r11)

m₀ is called the rest mass of the body.

Relativistic momentum: The relativistic momentum of a particle of rest mass m₀, moving with velocity v is defined as,

p = m_{0 g} v = m₀ g b c (r12)

where c is the velocity of light.

Einstein mass energy relation: A body with rest mass m₀, moving with velocity v, has its total energy e, given by,

E = m_0g c² = mc² (r13)

When the body is at rest its energy is given by,

E_rest = m₀c² (r14)

known as the rest mass energy. It follows from eq.(r13) and eq.(r14) the relativistic kinetic energy is given by,

K = m_0g c²- m₀c² = m₀c² (g -1) (r15)

Another important identity between the total energy, E momentum p and the rest mass of a particle is,

E² = p²c² + m₀²c⁴ (r16)

For small velocities v< < c, the equations (r7), (r10), (r12) and (r15) reduce to non relativistic equations of Galilean transformations, momentum and non relativistic kinetic energy.