THE POSTULATES OF SPECIAL RELATIVITY

(18) THE POSTULATES OF SPECIAL RELATIVITY

In 1905 A.Einstein gave 2 postulates that form the basis of his spec.th.of rel.: 1. The principle of relativity: The laws of physics are the same in all inertial reference frames. 2. The principle of constancy of the speed of light: speed of light in free space has the same value c in all inertial reference frames.

Both postulates are restricted to inertial frames. The first postulate declare that the laws of physics are universal and the same for all inertial observers. The 2^nd postulate was proved in CERN laboratory near Geneva. The experimenters measured directly the speed of the gamma rays emitted by the decaying pions which were moving with a speed of 0,99975c. According to Galileo, gamma rays should have a speed of c+0,99975c in the laboratory frame of reference, according to Einstein it should have c. The measured speed was equal to c within 0 %, which proves the 2^nd postulate. Both postulates imply that it is impossible to accelerate a particle to a speed greater than c.

RELATIVITY OF SIMULTANEITY.

Simultaneity: 2 events at different locations are simultaneous if an observer midway between them receives the flashes at the same instant. relativity of simultaneity: Spatially separated events are simultaneous in one frame are not simultaneous in another, moving relative to the first. This event is reciprocal.

TIME DILATION. A proper time T₀ is the time interval between 2 events as measured in the rest frame of a clock. In this frame both events occur at the same position. T₀= 2L₀/c. A light clock" - The time taken for light to travel from the source A' to the detector B' is 2L₀/c in the frame in which the clock is at rest. The emission and detection events occur at the same place in this frame. In a frame in which the clock is moving, the emission and detection events occur at 2 different positions. The time interval recorded is greater than recorded in the rest frame of the clock. T= g*T₀ where g=1/(1-v²/c²)^1/2,

Time dilation: 2 spatially separated clocks, A and B, record a greater time interval between 2 events than the proper time recorded by a single clock that moves from A to B 747i810h and is present in both events. The reality of time dilation was verified in an experiment performed by Rossi and Hall in 1941.

LENGTH CONTRACTION. The proper length L₀ of an object is the space interval between its ends measured in the rest frame of the object. L=1/g * L₀ Length contraction - The length measured in a frame moving relative to the rod is less than its proper length. Length contraction is reciprocal and only length parallel to the direction of motion are affected.

THE RELATIVISTIC DOPPLER EFFECT. In the classic Doppler effect for sound waves, the observed freq. depends differently on the v of the source and the observers. f_L = [(c-V)/(c+V)]^1/2 *f₀ . This equation applies when the source and observer move away from each other. When they approach, the sign of V is changed. Longitudinal effect when the signal travels along the direct. of motion. If the signals are detected perpendicular the transverse effect includes only the effect of time dilation T= g*T₀ or f_T = 1/T_T =1/g * f_0.

THE TWIN PARADOX. The paradox is resolved once we realise that the situation is not symmetrical. Twin B's turnover involves acceleration, so he switches from one inertial frame (if moving at a constant v) to another. Dt_B Dt_A g _.

We can explain it with the aid of effect of time dilation. B's clock will be delayed which is influenced by a potential field. After the journey, twin A will be older!

THE ADDITION OF VELOCITIES. The addition of any velocity to the speed of light yields the speed of light. This is specified in the second postulate. The speed of light is an absolute limit. The speed of material particle may approach but never equal c.

MASS - ENERGY EQUIVALENCE. Imagine an isolated box of length L with the light source at the one end and detector at the another. The mass of the box with the bulb and detector is M. When light waves transport an energy, they also transport linear momentum p E/c. Thus, if the source emits a pulse of light, the box will recoil with some velocity V. From conservation of linear momentum E/c= MV. If V<<c, it takes a time Dt =L/c for the pulse to reach the detector. When the light is absorbed, the box experiences an impulse that brings it to rest. In this time the box moves a small distance Dx = VDt = EL/Mc² . This is confusing, because we know from mechanics that no internal process can ever ever move the center of mass of the system. If we are to preserve this idea, we have to assume that the pulse transfers some mass m. from the source to the detector, through a displacement L. If the centre of mass is fixed, then (m.<<M.) -MDx +mL = 0. Taking both together m E/c² .If a body gives off the Energy in the form of radiation, its mass diminishes by E/c². The mass of a body is a measure of its energy content. The conservation of mass from mechanics should be replaced by the conservation of mass-energy. The equation E= mc² is the most significant outcome of special relativity.

(15) FISSION. Lisa Meitner and Otto Frish proposed a mechanism based on a "liquid-drop" model of nucleus put forward by Niels Bohr in 1936. In this model the nucleons are assumed to move about freely and randomly within the nucleus and to interact only with their nearest neighbours, like molecules in a drop. At the surface, the nucleons experience a net inward force. When a spherical uranium nucleus absorbs a neutron, it becomes unstable and undergoes oscillations. The shape of the "drop" can become distorted. If the neck forms, the short-range nuclear force between the 2 pairs of the "dumbbell" is greatly reduced. However, the long-range electrical repulsion between these 2 parts is only slightly diminished. This process was named fission (cell division). When a neutron is captured by ²³⁵₉₂U nucleus, it creates a short living compound nucleus ²³⁶₉₂U* in an excited state which then undergoes fission. n+²³⁵₉₂U->²³⁶₉₂U*->¹⁴⁰₅₄Xe+⁹⁴₃₈Sr+2n+Q . The final steps are several beta- and gamma decays to stable end products. The energy released in each fission event may be estimated as follows. The binding energy per nucleon of ranium is about 7 MeV whereas between A=90 and 150, its about 8,5 MeV. Thus the energy released in the fission process is about 200 MeV, which is many orders of magnitude greater thn the energies released in chemical reactions. About 170 MeVof energy is carried away as kinetic energy of the fission fragments: the rest is shared by neutrons emitted by the fragments , by beta particles, gamma rays, and neutrinos. The neutrons released in one fission event may be used to induce fission in other nuclei. Under suitable conditions, the process can repeat itself, thereby setting up a chain reaction. The energy is uncontrolled in nuclear bombs and controlled in nuclear reactors.

FUSION. The binding energy of light nuclei increases within atomic number. Therefore, when 2 light nuclei combine to form a larger nucleus, a process called fusion, energy is released. The energy released, per unit mass, is greater in a fusion reaction than in a fission reaction. In order for nuclei to fuse they must overcome the strong potential barrier created by their Coulomb repulsion. Consider 2 deuterons separated by 2* their radius. Their electrostatic pot. energy is U= ke²/r = 400 keV. Each deuteron would need a kin. energy of 200 keV to approach the other closely enough for the nuclear force to bind them. One way to give particles this much energy is to raise a gas to high temp. Fusion in the interior of the sun is the source of its energy. Energy in the sun is lower than 200 keV, but reaction occurs for 2 reasons. 1^st . there are always some particles at the tail of Maxwell distribution that have energies far greater than the average. 2^nd , particles can also tunnel through the Coulomb potential barrier. The vast amount of energy released in fusion is used in hydrogen bomb. Such a device, first exploded in 1952 involves an uncontrolled thermonuclear fusion reaction The necessary thermal energy to start the reaction is provided by the detonation of a fission bomb. In order to produce power from fusion reactions, 3 criteria must be met: 1. High temp. At such temp. atoms are stripped of their electrons, which results in a completely ionized gas called a plasma. 2. High particle density. This is required to increase the collision rate. 3. Long confinement time. Once particles have been brought together at high temp., they must be kept together long enough for the reaction to occur.

FISSION REACTORS. It is based on the process of fission of heavy nuclei. When a nuclei, such as ²³⁵₉₂U, undergoes fission, it releases neutrons that may be used to initiate fission in other nuclei, thereby creating a chain reaction. In a fission reactor, such a chain reaction must be fully controlled. Naturally occurring uranium consists of 0 % ²³⁵₉₂U and 99,3% ²³⁸₉₂U. When a ²³⁸₉₂U nucleus absorbs a neutron, it tends to emit a ray rather than undergo fission. In contrast, ²³⁵₉₂U has a high fission probab. for slow neutrons. The high energy neutrons produced in fission of ²³⁵₉₂U must be slowed down before they can induce further fissions. This is accomplished by a material called a moderator. The maximum transfer of kinetic energy from an incoming particle to a target particle occurs when when they have the same mass. Thus protons in water are ideal for this purpose. If the fuel is natural uranium, then the heavy water or graphite rods are the most efficient moderators, light water can be still used as moderator if the uranium is enriched by raising the proportion of ²³⁵₉₂U from 0,7% to 3-4%. An important parameter in chain reaction is the multiplication factor k .This is the ratio of the number of neutrons in one generation of the chain reaction to the number in the previous generation. When k=1, the number of neutrons produced is equal to the numb. which is absorbed or leak away. In this condition the system is said to be critical. In atomic bomb, two subcritical masses of uranium (enrichment 50%) are brought together to form a supercritical mass that explodes within 10^-8 sec. Since the enrichment in reactor fuel is much lower (<4%), a nuclear explosion cannot occur. However, when k>1 the thermal energy generated by the fission events and the radioactivity of the fission fragments can quickly melt the core, which can then melt the concrete below and so on (China syndrome). In addition, the moderating water would turn to steam and explode spreading radioactive material. In order to keep k close to 1, control rods of cadmium , which has a high absorption cross section for thermal neutrons, are inserted into the core. They can be dropped down or raised. In the pressurized water reactor core and moderating water are contained in the reactor vessel. The moderating water also serves as the coolant in the primary coolant system. In order to prevent the water from boiling very high pressure is required. The pipes in the primary system pass through a steam generator where water from secondary system is converted to high pressure steam and directed to a turbine, which is connected to an electrical generator. The 1 and 2 system are closed. Steam passes the turbine then cooled in condenser with water from lake, cooled in tower and discharge back to the lake.

FUSION REACTORS. J.D.Lawson deduced a necessary, but not sufficient condition for the net release of energy from fusion in a fusion reactor. If n is the particle density and t is the confinement time, then the Lawson criterion is (D-D) nt>10²² sec/m3, (D-T) nt>10²⁰ sec/m3. There are 2 basis approaches to confining a plasma to achieve the Lawson criterion. In the magnetic confinement technique a low particle density is compensated by a relatively long confinement time. In the system based on inertial confinement, the particle density is high but only for a short time. The Tokamak is a magnet. confinement device invented in USSR. The plasma in a Tokamak is confined by the combination of 2 magnetic fields. A strong toroidal field Bt is produced by 20 coils wrapped around the perimeter of a torus A weaker poloidal field Bp is produced by a large current that is induced in the plasma by a different time varying field generator by coils in the same plane as torus. The resultant magnetic field lines are helical and serve to confine the plasma. If the plasma were to come into contact with the walls of the containment chambers, the plasma would loose energy and cool down. The 14 MeV neutrons from the D-T reaction are absorbed by a molten lithium blanket surrounding the containment chamber. The thermal energy deposited in this blanket can then be used to produce steam for a conventional electrical generator. in the inertial confinement approach, the fuel is in the form of tiny pellets of diameter less than 1 mm, that contain a mixture of deuterium and tritium. The surface of pellets vaporises. As it expands, it sends a shock wave inward which increases the density of the core by a factor of and raises its temperature to over 10⁸ K. This occurs within 1.5 nsec, before the particles are able to disperse. That is they are confined by they own inertia. A continuous supply of power would be produced by fusing about 20-50 pellets each second. Features: Deuterium is easily extracted from sea water. Tritium is better because during a thermonuclear reaction involving tritium an emission of high energy neutrons may be avoided. A large amount of tritium is available on the Moon. A runaway reaction is not possible because of low quantity of fuel at any time. Radioactive wastes are less of a problem than with fission reactors.

(17) COSMOLOGY. THE EXPANSION OF THE UNIVERSE. Edwin Hubble deduced that the galaxies are moving away from one another and from us and that the greater is their distance from us the greater is their recessional speed. That is if d is the distance of the galaxy from the Earth and V is the speed with which the galaxy appears to be moving away from us Hubble's Law gives V=Hd where H is constant known as Hubble's parameter. The Hubble parameter has the dimension of inverse time. It's value can be learned only by experiment: we must deduce the distance of a galaxy from the Earth and its speed relative to Earth. he recessional speeds can be measured in a straightforward way using the Doppler shift of the light from the galaxy, but the distance scale is difficult to determine. Now H= 67km/sec*1/Mpc.

THE COSMIC BACKGROUND RADIATION. If galaxies are presently rushing apart, they must have been closer together in the distant past. If we run the cosmic clock back far enough, we find that in its early state the universe consisted of unimaginably high densities of matter and radiation. As the universe expanded both matter and radiation cooled; you can think of the wavelength of the radiant photons being stretched in the expansion. The radiation filled the entire universe in its compact state, and it continues to fill the entire universe in the expansion. We should still find that radiation present today, cooled to the extent that its most intense component is in the microwave region of spectrum. This is cosmic mic. b. rad. This radiation was first discovered in 1965 by Penzias and Wilson as annoying background hiss in during testing the antenna. Nobel in 1978. The thermal spectrum of radiation agrees with the theoretical curve representing the Planck spectrum for a source temp. of 2,735 K. The radiation has a uniform intensity in all directions, fills the entire universe uniformly, as would be expected gor the early universe. Finding temperature fluctuations in some regions of the sky by the COBE satellite was interpreted as the evidence for nonuniform distribution of matter in the early universe that led to the condensation of stars and galaxies.

THE BIG BANG COSMOLOGY. The universe began for unknown reasons some 10-20 billion years ago in a state of extreme density and temp .There where no galaxies or even clumped matter as we know it; the stuff of the universe at early times was a great variety of particles and antiparticles, plus radiation. The density of radiation and matter is related to the temp. of the universe. As the universe expands, it cools. The relationship between the temp. and the time after the formation of the universe: T= 1,5*10¹⁰s^1/2 K/t^1/2 . The radiation in the very early universe consisted of high energy photons whose typical energy can be roughly estimated as k_BT. The dominant process in the early universe can be represented as photons =particle + antiparticle. That is photons can engage in pair production and produce a particle-antiparticle pair. Conversely particle and antiparticle can annihilate into photons. In each case, the total energy of the photons must by at least as large as the rest of the energy. Formation and annihilation of of protons and neutrons: g+g<>p+p\ and g+g<>n+n\ where we represent the photons as the gamma rays. For photons to produce nucleon-antinucleon pair, the photon energy k_BT must be at least as large as the rest of the energy mc² of a nucleon. The minimum temp. of the universe that will permit production of nucleons and antinucleons is T=mc²/ k_B =1,09 * 10¹³ K. The universe cooled below this temperature at the time t=2ms. That is, at times earlier than 2ms the universe was hot enough for photons to produce nucleon- antinucleon pairs, but after 2ms the photons were not energetic enough to do it. But they still can produce electron-positron pairs!.

Early universe:

First 10^-32s. - The teory dealing with early stage of universe - inflation cosmology. Nature behaved for unknown reasons (dense matter) differently. Repulsive gravitation. Enermously rapid expansion of matter. Extension of size by factor of 10³⁰ during 10^-32s.

10^-32-10^-6s - T=1.5x10¹³K, strong interaction. Processes of combination of quarks, antiquarks into bayrons or mesons:

3q<->n V p  3q<->n^- V p^-

or collisions of bayrons to form mesons or new bayrons.

At time of 10^-6 universe is composed of protons, antiprotons, neutrons, antineutrons, mesons, antimes, leptons, antilep, photons. No of particles and their antipart are roughly equal. No of photons equal to no of protons, which is equal to no of electrons.

10^-6-10^-2s - particles and radiation have too little energy, so the era of strong interactions ends. Electromagn and weak-interact processes continue. Electromagn - production of particle-antiparticle pairs of lower rest energy, formation of protons and neutrons. Weak:

n+v_e<->p+e^- p+v_e^-<->n+e⁺

As long as leptons and neutrinos have enough energy, forward and reverse reactions rates are equal, so balance between no of charged leptons (e⁺, e^-) and neutrinos. Roughly equal no of protons and neutrons. Nucleon-antinucleon annihilation processes continue to occur.Up to 10^-2s still rough balance between no of protons and neutrons.

10^-2-1s - At t=1s nucleons consist of 73% protons and 27%neutrons. At 1s neutrinos have too little energy to cause proton-neutron transformations, but reverse continue:

p+e n+v_e n+e⁺->p+v_e^-

Time of neutrino decoupling - neutrinos continue to fill universe cooling along with its expansion.

1-6s - T=6x10⁹K, radiation cooloed to temp at which is not enough E to produce any particle-antiparticle pairs, so new particles are formed by pair production. Particle-antipart annihilation continues to occur. Electrons have too little E to cause protons ot transform into neutrons. The only weak interaction is decay of neutron:

n->p+e^-+v_e^-

Nucleons have 83%protons and 17% neutrons.

There are N protons, N electrons and 0.2N neutrons. Far more photons than nucleons or electrons. No of neutrinos=no of photons.

There is probably no antimatter now, because there was more matter than antimatter.

Big bang nucleosynthesis:

At first few seconds there were protons, neutrons, electrons, neutrinos, photons. Now there is mostly hydrogen and helium and some heavier elements. Formation of these elements is nucleosynthesis.

First step in building complex atoms is formation of deutron nuclei from neutron and proton:

n+p->d+Y binding E of d is 2.2MeV.

d+Y->n+p requires Y of 2.2MeV.

There are always some photons of E above 2.2

Formation of deutron nuclei begins at 280s. It reacts with p and n:

d+n->³H+Y d+p->³He+Y

H+p->⁴He+Y  ³He+n->⁴He+Y

Nearly al deutrons are converted into ⁴He.The temperature of universe when hydrogen atoms form is 3000K, after 700,000years. Once neutral atoms were formed, there are no free charged particles left. This is time of decoupling of matter and radiation field. Universe becomes transparent to radiation, which is observed as microwave background. Its temp is 2.735K now.