Introduction to special relativity Chapter 8 Chapter 1

Galilean-Newtonian ralativity Galilean relativity principle: the basic laws of mechanics are the same in all inertial reference frames. An inertial frame is one in which Newton's first law is valid. All inertial reference frames move with constant velocity with respect to one another. (a) (b) A person throws a ball from a moving car with velocity vball. a) In the reference frame of the car, the ball has velocity vball. b) In a reference frame fixed on the Earth, an observer measures the velocity of the ball to be vtot = vcar + vball.

Galilean-Newtonian ralativity From the Maxwell equations, the speed of light is constant and given by where ε0 and μ0 have the same values in all inertial reference frames. The invariance of c contradicts classic mechanics and Galilean relativity. Physicists thought that the speed of light was equal to c only in a particular propagation medium, which they called aether. According to classical mechanics the velocity of light seen by an observer should be vtot = c + vcar. This would mean that Maxwell’s equation were no longer valid. However, both the person on the truck and the observer on the ground measure the speed of the light as c, regardless of the speed of the truck. In the example, which is the reference frame in which light travels speed c?

Michelson-Morley experiment In order to resolve the apparent contradiction between Galilean relativity and Maxwell’s equations, Michelson and Morley designed an experiment to measure the speed of the Earth with respect to the aether (1883-1887). A beam of light is split by mirror A into two beams, which travel in perpendicular directions, and are reflected back to mirror A. • If the Earth is moving with velocity v relative to an aether: any difference in the times taken T1 and T2 for the beams to return to B would result in a phase difference and interference fringes in the recombined beam. • If there is no relative motion of the Earth to an aether: v = 0 and T1= T2 and hence the two beams would arrive back at A in phase and reinforce each other. No difference in phase between the two beams was observed – hence no relative motion.

postulates of the special theory of relativity A full explanation of the “null” result from the Michelson-Morley experiment was provided by Albert Einstein 18 years later: the aether could be dispensed with as no physical experiment can detect the absolute motion of an inertial reference frame. Einstein’s theory proposed that space is not three dimensional and Euclidean, but has the fourth dimension of time, and all these dimensions under go transformations between relatively moving reference frames. Principles of special relativity 1. The laws of physics are the same in all inertial reference frames. 2. Light propagates through empty space with speed c regardless of the relative speed between observers or between observer and the source. In summary, the principles of special relativity are valid for mechanics and electromagnetism, and the speed of light is the same in all reference frames.

simultaneity As time is not absolute, one of the implications of special relativity is that: observers, situated in distant places, do not necessarily agree on time intervals between events, or on whether they are simultaneous or not. Light comes from the two events at A and B, which are in the same rest frame. Observer O “sees” the lightning only when the light reaches O, the two events are simultaneous. If one observer O1 sees the events before the other O2, given that the speed of light is the same for each and has to travel different paths, the events do not appear to be simultaneous to them. Consider two observers who are moving with a relative velocity. The simultaneity of two events depends on the reference frames in which they are observed.

The Lorentz’s factor is time dilation Another thought experiment, using a clock consisting of a light beam and mirrors, shows that moving observers must disagree on the passage of time. The time it takes for light, travelling with the same speed, to travel across a moving spaceship with velocity v is longer for the observer on Earth than for the observer on the spaceship. There is an experience of time dilatation. The time interval in the moving frame (Δt) is related to the time interval in the rest frame (Δt0, called proper time) by: The Lorentz’s factor is and we can write

length contraction and velocity composition We found that time intervals are different in different reference frames. As a consequence, lengths must also be different. Length contraction is given by: or Length contraction occurs only along the direction of motion. Analogous with the proper time, the length of an object in its rest frame is defined as its proper length, which is equal to its length in its rest frame. As length shrinks and time expands, velocity also cannot have the same composition law as in the classical mechanics, the Lorentz transformation is used: If v << c, Lorentz transformation = Galilean transformation.

relativistic momentum The theory of special relativity means that momentum is also relativistic with magnitude defined as: Since the factor is always less than 1 and occurs in the denominator, the relativistic momentum is always larger than the non-relativistic momentum. The graph shows that for speeds attained by ordinary objects, such as cars and planes (< 0.2c), the relativistic and non-relativistic momenta are almost equal because their ratio is nearly 1. When the speed of an object is comparable to the speed of light, the relativistic momentum must be used, as it is significantly greater than the non-relativistic momentum. If v << c, Lorentz transformation = Galilean transformation. speed (v)

equivalence between mass and energy One of the most astonishing results of special relativity is that mass and energy are equivalent, in the sense that a gain or loss of mass can be equally regarded as a gain or loss of energy If an object of mass m travelling at a speed v, Einstein showed that the total energy E of the moving object is related to its mass and speed. When the object is at rest (v = 0 m/s), the total energy is called the rest energy E0: The rest energy represents the energy equivalent of the mass of an object at rest. The relationship between total energy and momentum is: One of the important consequences of the theory of special relativity is that objects with mass cannot reach the speed c of light in a vacuum. Thus, c represents the limit of speed.

learning the basics j j j j j 1. Physicists found experimentally that the speed of light was equal to c only in a particular medium of propagation. T F 2. Principles of special relativity are valid in mechanics and electromagnetism and the speed of light is the same in all inertial reference frames. T F 3. Observers, situated in distant places, do not necessarily agree on time intervals between events, or on whether they are simultaneous or not. T F j j j j j

applying the concepts 1. In the Michelson and Morley experiment, draw the paths of the two beams of light in the cases in which: the experiment moves with velocity v with respect to the aether; and there is no relative movement. Indicate also the time taken by light to travel the different paths.

applying the concepts 2. Draw the path for the light seen by an observer on the spaceship and on the Earth. Also, write the formula for the time taken by the light to travel for the two observers and indicate which observer measures the longer time.