SPECIAL THEORY OF RELATIVITY (CONT)

I : MASS AND ENERGY Einstein reworked Newton’s laws of mechanics using his newly discovered relativistic formulae. He found a formula for the energy of a moving object with mass m – Can now see what happens to the velocity of an object when we try to give it more and more energy.

What about objects moving at “small velocity”… it can be shown that: –The ½mv 2 is the Newtonian expression for the kinetic energy of a moving object. How slow is “slow”? –For car going at 30mph, approximate formula is wrong by 1 part in –For rocket going at 30,000mph, this approximate formula is wrong by 1 part in –So, this is right for all velocities experienced in everyday life.

Rest Mass Energy If we put V=0, we get… What does this mean? –Maybe its some fundamental “irreducible” (i.e., inaccessible) energy that every object possesses? –Maybe this energy can be accessed? In other words, maybe mass can be turned into energy? Turns out that this is correct. –Also, this can go the other way – energy can be turned into mass.

II : EXAMPLES OF CONVERTING MASS TO ENERGY Nuclear fission (e.g., of Uranium) –Nuclear Fission – the splitting up of atomic nuclei –E.g., Uranium-235 nuclei split into fragments when smashed by a moving neutron. One possible nuclear reaction is –Mass of fragments slightly less than mass of initial nucleus + neutron –That mass has been converted into energy (gamma-rays and kinetic energy of fragments)

Nuclear fusion (e.g. hydrogen) –Fusion – the sticking together of atomic nuclei –Much more important for Astronomy than fission e.g. power source for stars such as the Sun. Explosive mechanism for particular kind of supernova –Important example – hydrogen fusion. Ram together 4 hydrogen nuclei to form helium nucleus Spits out couple of “positrons” and “neutrinos” in process

–Mass of final helium nucleus plus positrons and neutrinos is less than original 4 hydrogen nuclei –Mass has been converted into energy (gamma-rays and kinetic energy of final particles) This (and other very similar) nuclear reaction is the energy source for… –Hydrogen Bombs (about 1kg of mass converted into energy gives 20 Megaton bomb) –The Sun (about 4  10 9 kg converted into energy per second)

Anti-matter –For every kind of particle, there is an antiparticle… Electron  anti-electron (also called positron) Proton  anti-proton Neutron anti-neutron –Anti-particles have opposite properties than the corresponding particles (e.g., opposite charge)… but exactly same mass. –When a particle and its antiparticle meet, they can completely annihilate each other… all of their mass is turned into energy (gamma-rays)!

III: EXAMPLES OF CONVERTING ENERGY TO MASS Particle/anti-particle production –Opposite process to that just discussed! –Energy (e.g., gamma-rays) can produce particle/anti- particle pairs –Very fundamental process in Nature… shall see later that this process, operating in early universe, is responsible for all of the mass that we see today!

Particle production in a particle accelerator –Can reproduce conditions similar to early universe in modern particle accelerators…

CERN

A real particle creation event

Spacetime diagram and motion Changing from one reference frame to another… –Affects time coodinate (time-dilation) –Affects space coordinate (length contraction) –Leads to a distortion of the space-time diagram as shown in figure.  

II: SIMULTANEITY Consider an observer in a room. Suppose there is a flash bulb exactly in the middle of the room. Suppose sensors on the walls record when the light rays hit the walls. Since speed of light is constant, light rays will hit opposite walls at precisely the same time. Call these events A and B.

Imagine performing same experiment in a room aboard a moving spacecraft (observed by someone at rest). The light rays will not strike the walls at the same time (since the walls are moving!). Event A will happen before event B. But astronaut in spacecraft thinks events are simultaneous. Concept of “events being simultaneous” (i.e. simultaneity) is different for different observers (Relativity of simultaneity).

The order of events and causality Consider same experiment seen by three observers –Astronaut thinks events A and B are simultaneous –Observer at rest thinks A occurs before B

–What about a 3 rd observer who is moving faster than spacecraft? –3 rd observer sees event B before event A –So, order in which events happen depend on frame of reference.

t A3 t B3 t A2 t B2 ct 2 ct 3 AB t A1 t B1 ct 1 x Light cone t A1 =t B1 t A2 <t B2 t A3 >t B3

The laser gun experiment Suppose there is a laser gun at one end of spacecraft, targeted at a victim at the other end. Laser gun fires (event A) and then victim get hit (event B). Can we change the order of these events by changing the frame of reference? i.e., can the victim get hit before the gun fires?

These events are causally-connected (i.e. one event can, and does, affect the other event). In fact, it is not possible to change the order of these events. This is true provided that –The laser bolt does not travel faster than the speed of light –We do not change to a frame of reference that is going faster than the speed of light

III : SPACE-TIME DIAGRAMS “Light Cone”

Causality Events A and B… –Cannot change order of A and B by changing frames of reference. –A can also communicate information to B by sending a signal at, or less than, the speed of light. –This means that A and B are causally-connected. Events A and C –Can change the order of A and C by changing frame of reference. –Any communication between A and C must happen at a speed faster than the speed of light. If idea of cause and effect is to have any meaning, we must conclude that no communication can occur at a speed faster than the speed of light.

Past, future and “elsewhere”. “Future of A” (causally- connected) “Past of A” (causally- connected) “Elsewhere” (causally- discounted)