Consequences of Special Relativity Simultaneity: Newton’s mechanics ”a universal time scale exists that is the same for all observers” Einstein: “No universal time scale exists that is the same for all observers. A time interval measurement depends on the reference frame in which measurement is made”. Einstein Thought Experiment: Two lightning bolts strike the ends of a boxcar, leaving marks on the boxcar and the ground. Which observer is correct? Both. There is no preferred inertial frame. Simultaneity is not absolute.

Time dilation Observers in different inertial frames measure different time intervals. Thought Experiment: Moving vehicle. Proper time: the time interval between two events as measured by an observer who sees the events occur at the same point in space (in his frame). A moving clock runs slower than a clock at rest by a factor gamma. Generalization: All physical processes, including chemical reactions and biological processes slow down relative to those of a stationary object when they occur in a moving frame. Example: Heartbeat of an astronaut. Experiment: Decaying muons (unstable particles)

Length Contraction Thought Experiment: A spaceship moving from one star to another. Two observers, one on Earth (stationary) and one on the spaceship. measured distances between two objects depend on the frame on which the measurement is made. Proper length: when measurement is made by someone who is at rest with respect to the object. The observer on the other frame will always measure contracted length. If an object has a length L’ when at rest, then an observer moving with respect to him with relative speed v, in a direction parallel to its length, it will appear contracted. Experiment: Decay of mouns

Lorentz Transformation Michelson-Morley Experiment: No ether, and Galilean transformations are not valid when v approaches c. Need correct (general) transformation for all speeds. Lorentz Transformation: a set of formulas that relates space and time coordinates of inertial observers moving with a relative speed. Consider the motion of a rocket (S’ frame) relative to S frame with speed v along xx’ axis. Find Lorenzt transformation between coordinates (x’,y’,z’,t’) and (x,y,z,t). Lorentz velocity transformation. Time dilation and length contraction are contained in Lorentz transformation.

Relativistic Form of Newton’s Laws We need to generalize Newton’s equations to conform to Lorentz transformation and relativity. The generalization should reduce to classical definitions for v<<c. Conservation of momentum states that when two bodies collide, the total momentum remains constant. Suppose a collision is described in reference S in which the momentum is conserved. If velocities in frame S’ (moving) are calculated using the Lorentz transformation, and the classical definition P=mv is used, we will find that the momentum in S’ is not conserved. This contradicts the postulate of relativity. We need to modify the definition of momentum Force will also be modified Energy will also be modified.

Mass as a measure for energy From the mass-energy equation: even if a particle is at rest, it posses enormous energy through its mass. This is clear in Nuclear (Radioactive) and particle physics interactions: Conversion of energy into mass and conversion of mass into energy. Classical laws separate conservation of energy law from conservation of mass law. Special Relativity: Conservation of mass-energy: the sum of mass-energy of a system of particles before interaction must equal the sum of mass- energy of the system after interaction. Example: Inelastic collision of two particles. Kinetic energy is not lost in an inelastic collision but shows up as an increase in the mass of the final composite object. Both relativistic mass-energy and momentum are always conserved in a collision Fission: Decay of large radioactive nuclei