1 Relativity H4: Some consequences of special relativity

2 Twin paradox  An interesting and amusing result predicted by relativity theory is often called the twins "paradox".  If one of a pair of twins goes on a long, fast journey and then returns home, it will be found that the twins have aged differently  the "stay-at-home" twin is always the older

3 Twin paradox  The special theory of relativity deals only with inertial frames of reference, the astronaut twin would have undergone huge accelerations, thus his frame is not inertial

4 Hafele-Keating experiment  This was an experiment to demonstration the twin paradox.  Two atomic clocks were flown around the world in opposite directions.

5 Hafele-Keating experiment  The clocks were compare with a stationary clock. Eastward Journey Westward Journey Predicted -40 +/- 23 ns /- 21 ns Measured -59 +/- 10 ns /- 7 ns

6 Hafele-Keating experiment  This experiment showed that the clocks had experienced time dilation, compare to the stationary clock.  One clock had slowed down and one clock had sped up

7 Vector addition  At normal speeds, you just add the velocities, at relativistic speeds the correct formula is  if ‘u’ is in the same direction as ‘v’

8 Mass and energy  The mass of a body is a relative concept.  The mass of a body measured by an observer at rest relative to the body is called (not surprisingly) the rest mass of the body.  The mass (sometimes called the relativistic mass) of the body measured by other observers depends on the velocity of the observer relative to the body.

9 Mass and energy  As the variation of mass is basically due to the time dilation effect, you should not be surprised to find that, if the rest mass of a body is mo, then its mass, m, as measured by an observer moving with speed v relative to the body is given by

10 Mass and energy  The Einstein relationships are:  E = mc 2 (total energy)  E = m 0 c 2 (rest energy)

11 Mass and energy  If a force causes body B to accelerate away from observer A (as in the example above) then work is done by that force.  As usual we can define the kinetic energy possessed by body B (as measured by A) by saying that

12 Mass and energy  K.E. of B = Work Done causing it to accelerate  It can be shown that if the relative speed of body B is such that its mass (as measured by A) is m, then  K.E. = mc 2 - m o c 2 or K.E. = (change in mass)c 2

13 Mass and energy  Here we are seeing the equivalence of mass and energy. Some of the work done by the force is converted into mass and if we define the total energy, E, possessed by a body to be the sum of its rest energy (m o c 2 ) and its K.E. we have the famous result  E =  mc 2

14 Mass and energy  It is easy to show that if the velocity of the body relative to the observer is small compared with the velocity of light, then the relativistic formula reduces to the Newtonian expression (½mv2)  If the velocity of the body relative to the observer is very close to the velocity of light, virtually all the work done by the force is converted to mass.  Experiments involving high speed particles (protons, electrons etc.) in particle accelerators give evidence to support the idea that mass varies with velocity.

15 Units of Mass/Energy  The S.I. unit for mass is the kg and for energy, the Joule. However, on the scale of subatomic particles, we often use MeV for energy (1MeV = 1·6× J). So, a possible unit for the quantity "mc 2 " is MeV.  For this reason, the masses of subatomic particles are often expressed in MeV/c 2.  For example the rest energy of an electron is about 511MeV and its mass is therefore said to be 511MeV/c 2.