\relativ.htm(Ó R. Egerton)

Einstein's Special and General Theories of Relativity

In 1905, Albert Einstein published a paper which revolutionized our thinking about space and time and which he (and others) subsequently developed into the **Special Theory** of Relativity. This theory describes how physical properties with which we are familiar (mass, length, period of oscillation of a physical system, etc) would appear if viewed by an observer who is in **uniform motion** (constant velocity) relative to the observed object.

Einstein later introduced his **General Theory**, which allows for nonuniform motion (presence of an **acceleration**) and gives an explanation of the force of **gravity**. Combined with ideas of quantum physics, this theory led to the prediction that certain properties of light would be affected by gravitational fields, and to prediction of black holes for example.

SPECIAL RELATIVITY

Central to the discussion of special relativity is the idea of an **inertial frame** (or reference). This is basically a coordinate system, which might be attached to an observed object or to the observer, which undergoes no acceleration. Consequently, the **relative velocity** between two inertial frames is necessarily constant, providing what we refer to as **uniform motion**.

Einstein based his 1905 theory on two postulates:

1. No physical measurement can distinguish one inertial frame from another.

2. The speed of light (in vacuum) is the same in all inertial frames, regardless of any motion of the source.

Postulate (1) is also known as the **Principle of Relativity** and is a generalization of the idea of Galileo: that uniform motion is undetectable by *mechanical* experiments. This **Galilean Principle of Relativity** accounts for the fact that there are no obvious effects of the earth's motion through space, as it orbits the sun (at a tangential speed of about 17 000 km/hr !). For example, objects released from the top of a tower fall vertically downwards (towards the centre of the earth), as they would if the earth were stationary and not at some angle which depends on the earth's tangential speed. Einstein thought that Galileo's principle should apply to the whole of physics, including electromagnetic phenomena.

Postulate (2) derives from the idea of Maxwell that light behaves as a travelling wave, containing oscillating electric and magnetic fields which can advance (propagate) in a vacuum at a speed, denoted by the symbol c , which depends only on two basic constants of electrostatic and magnetic theory. The fields require no medium for their existence, unlike the case of sound waves (for example) whose velocity within a gas (or liquid or solid) *is* affected by any motion of this medium.

Previously, physicists had assumed that light waves behave somewhat like sound, propagating though an invisible medium which they termed the (luminiferous) **ether**. Attempts to measure the speed of the earth relative to the ether failed; in particular, the Michelson-Morley experiment (performed repeatedly between 1881 and 1930) showed that light travels with *exactly* the same speed in two perpendicular directions, which is impossible if the earth is moving (due to its orbit around the sun) through an ether.

Length contraction

From Einstein's two postulates, several properties follow *as a matter of pure logic*. One of these is an effect called **length contraction**: the measured length (in the direction of motion) of an object which is moving at uniform speed v *relative to* an observer is less than if the object were stationary. The length measured when there is *no* relative motion is called the **proper length** and all other lengths are called improper. The length-contraction effect can be expressed mathematically as:

improper length = (1/gamma) (proper length)

where 1/gamma = (1 - v^2 / c^2)^(1/2) and is *less than* unity; here ^ means* to the power of* , so ^(1/2) means taking the *square root*. Since the value of c (= 3.00 x 10^8 m/s) is so large, length contraction is entirely negligible (e.g. 1 part in 2 x 10^12 for v = 1000 km/hr) for objects such a cars, trains and airplanes.

There is no change in dimensions of the object which are *perpendicular* to the relative velocity v ; therefore it might be expected that a fast-moving cube would appear squashed (in the direction of motion) in a high-speed photograph. However light from different parts of the cube takes different times to reach the camera, so the photograph is *not* a record of the object at a single instant of time. This illustrates the difference between a true **measurement** and a simple **observation**. In fact, the cube would appear as if it had been rotated (through a fixed angle), due to the combined effects of length contraction and the finite (limited) speed of light.

**Magnetic force** can be thought to arise from electrostatic interaction, plus the length-contraction effect. For example, a metal contains potentially-mobile negative electrons and an equal number of immobile positive charges. In the absence of any electrical current, two parallel wires exert no force on each other because the attractive forces (electrons in one wire attracting positive charge in the other, and vice versa) are exactly balanced by repulsions (electrons in one wire repelling those in the other, likewise for the positive charges). With an equal current travelling in the same direction in each wire, the repulsive forces are unchanged (there is no relative motion between the electrons or positive charges) but the attractions are increased, since the positive charge "sees" the distance between the moving electrons as contracted (equivalent to an increase in negative charge per unit length of the wire) or vice versa. This increase is seen as a net attractive force between the two wires, usually attributed to the magnetic effect of the currents. We can say that Special Relativity unites the concepts of magnetic and electrostatic force into a single *electromagnetic* force.

Time dilation

Another effect predicted by special relativity is **time dilation** : a clock moving at uniform speed relative to an observer would be measured to run slow, arising from the properties of space-time and not from the finite speed of light. By analogy with the above, we can define an interval of proper time as a difference in the readings of a clock which is *stationary* with respect to the observer; where there is relative motion, we measure an improper time interval. Analysis of a simple situation (Casper and Noer pp. 333 - 337) shows that

improper time interval = (gamma) (proper time interval)

Since gamma > 1, the interval between ticks of a "moving" (relative to the observer) clock is greater than for a "stationary" clock, so "a moving clock runs slow". This effect has been verified by carrying highly-accurate atomic clocks aboard aircraft and comparing their "readings" with those of an identical clock which remained stationary. Although the difference in elapsed time is miniscule, the extremely high accuracy of the atomic clock has allowed the time dilation effect predicted by Special Relativity to be verified.

A more extreme (but hypothetical) example is the case of two twins: one remains on earth, the other journeys at a high speed (approaching the speed of light) to a distant star and back. Upon returning, the moving twin will have aged less than the twin who stayed on earth. Although this is not a simple situation, since accelerations are necessarily involved in the return journey, detailed analysis shows that Special Relativity gives the right answer for the difference in age.

Relativistic mass

Later in 1905, Einstein published a paper which shows that Newton's second law (F = ma) applies to any object, travelling at any speed v, provided its usual mass (called the rest mass, if measured when the object is stationary) is replaced by a **relativistic mass** given by:

relativistic mass = (gamma) (rest mass)

Since gamma > 1, there is a relativistic **increase** in mass. Therefore, if a constant force F is applied to a stationary object, it initially accelerates at a constant rate a = F/m0 (where m0 is its rest mass) but as the speed v approaches c , gamma becomes significantly larger than unity, the relativistic mass m significantly exceeds m0 and the rate of acceleration (a = F/m) decreases. In fact, the acceleration tends towards zero as v approaches c : no material object can travel at or above the speed of light (in vacuum). At high speeds, the work done by the force F goes into increasing the relativistic mass, rather than the speed. In other words, energy provided by the force is *converted into mass*. Einstein introduced the concept of the total energy E of an object

E = m c^2 = (gamma) m0 c^2 = K + m0 c^2

as being the sum of its kinetic energy K and its **rest energy** E0 = m0 c^2 . From this equation, it is easy to show that the correct general formula for kinetic energy is:

K = (gamma - 1) m0 c^2

rather than the classical expression: K = (1/2) m0 c^2 . However, Einstein's general formula is *consistent* with the classical expression, since for v<<c we can use the **binomial theorem**:

(1+x)^n = 1 + n x + (1/2)n(n-1) x^2 + ... = 1 + n x (approximately) if x<<1

with x = -v^2/c^2 and n = -1/2 , so that gamma = (1+x)^n , giving

K = (1 + nx - 1) m0 c^2 = (nx) m0 c^2 = (-1/2) (-v^2/c^2) m0 c^2 = (1/2) mo v^2

For v<<c, Special Relativity gives the same result as Classical Physics, an example of the **Correspondence Principle** which states that a new scientific theory must give the same predictions as an older theory under conditions in which the older theory has already been found to be correct.

One situation in which speeds comparable to c are routinely achieved is in the acceleration of charged particles, for example electrons in a TV tube, oscilloscope or electron microscope, or other particles in a nuclear accelerator. The particles (charge q) may be accelerated via an electrostatic field, by applying a potential U to an accelerating electrode. The gain in kinetic energy of a particle is then equal to its loss in potential energy: K = - q U . For the electrons in color-TV tube, q = -e = -1.6E-19 Coulomb, m0 = 9.11E-31 kg and U = +20,000 volts, giving K = 3.2E-15 Joules = (gamma - 1) m0 c^2, so that gamma = 1.04 and v = 0.27 c . The electron-optic design of the tube must take into account the relativistic mass increase, and the fact that such designs work as expected is further evidence for the the accuracy of the Special Theory.

The conversion of energy into mass leads to the concept that mass and energy are somewhat equivalent, and that it is actually the *total* which is conserved. The **conservation of mass-energy** therefore replaces our previous idea of the separate conservation of these two quantities. The reverse process, conversion of mass into energy, is also possible. For example, a star like our sun converts hydrogen into helium and heavier elements within its core, via a *nuclear* reaction. The products of this reaction have slightly less mass than the original hydrogen and this difference in mass (Dm) accounts for the radiant energy (E = Dm c^2 = 4.0E16 J/s) liberated. The same principle applies to nuclear reactors and nuclear weapons, and even to the energy liberated in chemical reactions (where the change in mass is too small to be measured).

GENERAL RELATIVITY

In the General Theory, Einstein concerned himself non-inertial frames, in other words motion which involves acceleration. One example is uniform *circular* motion, where the tangential speed v of an object (in a circle of radius R) is constant but its direction of this velocity is continually changing. According to Newtonian mechanics, this change in velocity is equivalent to a centripetal acceleration v^2/R towards the centre of the circle, and requires an inward centipetal force F = mv^2/R , otherwise the object flies off at a tangent (reverts to uniform *linear* motion).

However, the situation can also be analysed from the point of view of an observer in the non-inertial reference frame which rotates *with* the object. To such an observer, there appears to be a **centrifugal** force causing objects to move outward, away from the centre of rotation. In addition, an object which is projected at a uniform speed and which moves in a straight line, according to non-rotating observers, moves in a curved path in the non-inertial frame - evidence of a **Coriolis** force in the rotating coordinate system. The Coriolis and centrifugal forces are referred to a **fictitious forces**; they appear as two extra terms if Newton's laws of motion are written in terms of coordinates measured with respect to a rotating set of axes.

Strictly speaking, the earth is a non-inertial platform: it rotates about the sun, as well about its own polar axis. Direct evidence of this rotation comes from the behaviour of a simple pendulum with a large mass, such that its motion continues for many hours (called a Foucault pendulum). The direction of swing appears to rotate (precess) through 360 degrees every 24 hours, for a pendulum is located at the north or south pole. This can be understood as the earth rotating "beneath" the pendulum, or as the effect of a sideways Coriolis force acting perpendicular to the direction of the swing. The rate of precession is less at lower latitudes and becomes zero at the equator.

The existence of these fictitious forces apparently provides a means of detecting a non-inertial frame. In other words, accelerated motion appears have an *absolute* existence, whereas linear motion is always* relative* (to another object). Another example is provided by Newton's water bucket experiment (C&N p. 403). If the bucket and its contents are rotating at the end of a long rope, the water surface will become curved, due to the centifugal force (according to ab observer rotating with the bucket). Observation of this curvature provides direct evidence that the system is rotating, without reference to any external object. However, Newton's explanation was rejected on philosophical grounds by the German philosopher Ernst Mach (1838-1916), who maintained that motion of any kind (linear or rotational) can have no real existence unless specified with respect to other objects. So what other objects define the rotation of a water bucket and cause the water surface to become curved ? According to Mach, the rotation must be defined relative to the rest of the universe, including stars and galaxies at immense distances which together exert a force on the water. Satellite experiments designed to detect such a force, arising from the acceleration of one object relative to another, have been partially successful.

The Principle of Equivalence

Mach interpreted the centrifugal force introduced by rotational motion as a real force. Einstein decided to invert this logic and describe "real" forces as if they were fictitious, arising from the choice of non-inertial reference frame. He applied this interpretation to the gravitational force, so his General Theory can be regarded as an explanation of gravity.

A simple example is that of an artificial satellite or space capsule, orbiting the earth in a circle. Inhabitants of the capsule feel weightless: gravity has become zero in the non-inertial frame which rotates with the capsule. In the language of General Relativity, they are following circular contours of space-time which represent the gravitational field of the earth. Another example is a free-falling elevator: the inhabitants feel weightless because the earth's gravitational field becomes zero in the accelerating frame of the elevator

Einstein embodied this close connection between gravity and acceleration in his **Principle of Equivalence**: no physical measurement can distinguish a reference frame which is accelerating (relative to an inertial frame) from one which is placed in a uniform gravitational field. Therefore, *no* experiment inside an elevator can distinguish whether reaction forces (which provide the impression of eight) arise from gravity, acceleration or a mixture of the two.

If the elevator contains a light source which projects a beam perpendicular to the direction of motion, this beam must appear straight according to external observers (if the elevator has glass walls) and would therefore appear slightly curved according to its occupants, who are accelerating pependicular to its path. We could interpret this curvature as being due to acceleration, but according to the Principle of Equivalence it could also be the effect of a local gravitational field.

Einstein had already proposed (in 1905!) that light can be regarded as a stream of particles, now known as photons. Therefore, curving of a light beam by gravity is evidence that photons have a **gravitational mass**. Experiments have confirmed this. For example, the positions of stars seen near the disk of the sun appear slightly shifted in position during a total eclipse, due to bending of the starlight as it passes through the strong gravitational field close to the surface of the sun.

Also, light from a massive star suffers a **gravitational redshift**: the photons lose internal energy as they do work against the gravitational field, and this can be measured as a decrease in frequency of the light, or an increase in wavelength towards the red end of the spectrum. An extreme case is a star which has collapsed to very small dimensions and whose local gravitational field is so strong that it prevents photons from leaving. Such a **black hole** emits no radiation and so cannot be seen directly. However its gravitational field may act a lens which deflects light from more distant stars, sometimes producing a double image. If the black hole is close to a visible star, the two objects may rotate about their common centre of mass and observation of the visible star may provide evidence of its dark companion. Also, ionized interstellar gas which is gravitationally attracted and accelerated into a black hole can emit electromagnetic radiation. The x-ray source Cygnus-X1 (which has a power output of 10^30 Watt) is believe to be an example.