Time Dilation, Summary The time interval Δt between two events measured by an observer moving with respect to a clock is longer than the time interval Δtp between the same two events measured by an observer at rest with respect to the clock A clock moving past an observer at speed v runs more slowly than an identical clock at rest with respect to the observer by a factor of

Identifying Proper Time The time interval Δtp is called the proper time The proper time is the time interval between events as measured by an observer who sees the events occur at the same position You must be able to correctly identify the observer who measures the proper time interval

Time Dilation – Generalization All physical processes slow down relative to a clock when those processes occur in a frame moving with respect to the clock These processes can be chemical and biological as well as physical Time dilation is a very real phenomena that has been verified by various experiments

Time Dilation Verification – Muon Decays Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron Muons have a half-life of Δtp = 2.2µs when measured in a reference frame at rest with respect to them (a) Relative to an observer on earth, muons should have a longer lifetime (b) A CERN experiment measured lifetimes in agreement with the predictions of relativity

Redefinition of Momentum Modification of Newton’s Law Only one modification is necessary to make Newton’s mechanics To be consistent with the Principle of Relativity. Redefinition of Momentum mo is called the rest mass, its mass when the object is at rest. We can also define the relativistic mass m

Then, the relativistic form of Newtonian mechanics reads: In the absence of forces, a body moves with uniform speed in a straight line. 2. In the presence of force, the body changes its state of motion in such a way that Mass (inertia) depends on the speed of the object. In the limit of v/c << 1, the relativistic mechanics recovers the old form!!

speed (m/s) rel. mass (kg) Note 100 Rest mass 90 NASCAR 1,000 MACH 3 32,000 100.0000006 Planetary speed 1000,000 100.0006 0.98c 502.5 Decent accelerator

m/mo = 1/sqrt(1 – v2/c2) (m/mo)2 = (1.010)2 = 1.020 = 1/(1 – v2/c2) Ex. 29.1. How fast must an object be moving if m/mo = 1.010? m/mo = 1/sqrt(1 – v2/c2) (m/mo)2 = (1.010)2 = 1.020 = 1/(1 – v2/c2) v2/c2 = 0.02/1.020  v = 0.14c = 4.2 x 107 m/s We have to apply force to make an object move faster. However, as the speed of an object gets larger, its mass gets larger. Therefore, we have to apply larger force to increase its speed at the same rate. It is impossible, from the point of view of the relativistic equations, to Accelerate a particle to a speed larger than that of light!!

How about energy and work in the relativistic world? Modification is needed accordingly to be consistent with the new definition of momentum. Kinetic energy = E (total energy without force) Relativistic case

the most famous equation in physics!! Relativistic case approximate for small v typical kinetic energy For a particle at rest, the most famous equation in physics!!

“If a body gives off the energy E in the form of radiation, its mass Meaning of “If a body gives off the energy E in the form of radiation, its mass diminishes by E/c2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that the mass of a body is a measure of its energy-content; if the energy changes by E, the mass changes the same sense by E/(9 x 1020), the energy being measured in ergs, and mass in grams.” A. Einstein in The Principle of Relativity, Methuen, London, 1923. It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salt) the theory may be successfully put to test

mo = 1 kg E = moc2 = (1 kg) x (3 x 108 m/s)2 = 9 x 1016 J Ex. 29.2 A kilogram of gasoline (about 1.5 liters) yield an energy of 4.8 x 107 J when burned. Compare this to the mass energy of 1 kg Of a substance. mo = 1 kg E = moc2 = (1 kg) x (3 x 108 m/s)2 = 9 x 1016 J Compare this with 4.8 x 107 J This huge difference stresses the importance of nuclear energy.