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Galilean Transformation. S. S’. (x,y,z) and (x’,y’,z’) are the coordinates of the same point measured respectively in S and S’. S and S’ are inertial frames. u’, a’ = velocity and acceleration as measured in moving frame. u’, a’. u, a = velocity and acceleration as measured in fixed frame.
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Galilean Transformation S S’ (x,y,z) and (x’,y’,z’) are the coordinates of the same point measured respectively in S and S’ S and S’ are inertial frames
u’, a’ = velocity and acceleration as measured in moving frame u’, a’ u, a = velocity and acceleration as measured in fixed frame According to Galilean transformation:u’ = u – v a’ = a All equations of Classical Mechanics areinvariantunder Galilean transformation. The laws of physics are the same in all inertial frames. Maxwell equationsare not invariant under Galilean transformation: c is a universal constant The velocity of light is independent of source or detector velocity (Michelson-Morley experiment, double star images, aberration of star positions, ...)
Speed of Light Experimental measurements of the speed of light have been refined in progressively more accurate experiments since the seventeenth century. Recent experiments give a speed of but the uncertainties in this value are chiefly those of comparisons to previous standards for the length of the meter. Therefore the above speed of light has been adopted as a standard value and the length of the meter is redefined to be consistent with this value. The speed of light in a medium is related to the electric and magnetic properties of the medium, and the speed of light in vacuum can be expressed as
Maxwell's Equations Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships. ρ, J = charge, current density
Lorentz Transformation The primed frame moves with velocity v in the x direction with respect to the fixed reference frame. The reference frames coincide at t=t'=0. The point x' is moving with the primed frame. The reverse transformation is: Much of the literature of relativity uses the symbols β and γ as defined here to simplify the writing of relativistic relationships. Maxwell equationsare invariant under Lorentz Transformation
S S’ P(x’, y’, z’) in moving frame P(x, y, z) in fixed frame Lorentz Transformation implies: c2 t2 – (x2 + y2 + z2)= c2 t’2 – (x’2 + y’2 + z’2) (Lorentz invariant) x’2 + y’2 + z’2 x2 + y2 + z2 c2 is the same (speed of light )2 in the two frames = = t’2 t2 t and t’ cannot be equal Two “events” (x1, t1) and (x2, t2) at the same time t1 = t2 in S are do not happen at the same time in S’, t1’ ≠ t2’.
Length Contraction The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation. Thelength is maximum in the frame in which the object is at rest.
Time Dilation A clock in a moving frame will be seen to be running slow, or "dilated" according to the Lorentz transformation. The time will always be shortest as measured in its rest frame. The time measured in the frame in which the clock is at rest is called the "proper time". The time measurementsmade in the moving frame are made at the same location, so the expression reduces to: For small velocities at which the relativity factor is very close to 1, then the time dilation can be expanded in a binomial expansion to get the approximate expression:
Muon Experiment – Earth observer Muon and Earth observer agree on the number of muons which reach the ground
Twin Paradox The story is that one of a pair of twins leaves on a high speed space journey during which he travels at a large fraction of the speed of light while the other remains on the Earth. Because of time dilation, time is running more slowly in the spacecraft as seen by the earthbound twin and the traveling twin will find that the earthbound twin will be older upon return from the journey. The common question: Is this real? Would one twin really be younger? The basic question about whether time dilation is real is settled by the muon experiment. The clear implication is that the traveling twin would indeed be younger, but the scenario is complicated by the fact that the traveling twin must be accelerated up to traveling speed, turned around, and decelerated again upon return to Earth. Accelerations are outside the realm of special relativity and require general relativity. Despite the experimental difficulties, an experiment on a commercial airline confirms the existence of a time difference between ground observers and a reference frame moving with respect to them.
The Einstein velocity relationship transforms a measured velocity as seen in one inertial frame of reference (u) to the velocity as measured in a frame moving at velocity v with respect to it (u'). In problems involving more than two objects, the main difficulty is the assignment of velocity to all the objects. If A sees B moving at velocity v, then a velocity measured by B (u') would be seen by A as: These relationships make perfect sense at low speeds where both denominators approach 1. If v = u’ = 0.9 c one has u = 1.8 c/1.81 < c If v or u’ = c, then also u =c
Just taking the differentials of these quantities leads to the velocity transformation. Taking the differentials of the Lorentz transformation expressions for x' and t' above gives with the reverse transformation
Relativistic dynamics and mass Special relativity leads to an increase in the effective mass of a particle with speed v, given by the expression (relativistic mass) It follows from the Lorentz transformation when collisions are described from a fixed and moving reference frame, and it arises as a result of conservation of momentum. The increase in relativistic effective mass makes the speed of lightc the speed limit of the universe. This increased effective mass is evident in cyclotrons and other accelerators where the speed approaches c. Exploring the calculation above will show that you have to reach 14% of the speed of light, or about 42 million m/s before you change the mass by 1%. The speed of light c is said to be the speed limit of the universe because nothing can be accelerated to the speed of light with respect to you. A common way of describing this situation is to say that as an object approaches the speed of light, its mass increases and more force must be exerted to produce a given acceleration. There are difficulties with the "changing mass" perspective, and it is generally preferrable to say that the relativistic momentum and relativistic energy approach infinity at the speed of light. Since the net applied force is equal to the rate of change of momentum and the work done is equal to the change in energy, it would take an infinite time and an infinite amount of work to accelerate an object to the speed of light.
Mass, energy and momentum The formulation of dynamics in Special Relativity leads to the energy-mass relationship moc2 = E = mc2 includes both the kinetic energy and rest mass energy for a particle. The kinetic energy T of a high speed particle can be calculated from Notice that, for small velocities The relativistic momentum of a particle is given by
Relativistic Energy in Terms of Momentum The famous Einstein relationship for energy can be blended with the relativistic momentum expression to give an alternative expression for energy. The combination pc shows up often in relativistic mechanics. It can be manipulated as follows: by adding and subtracting a term it can be put in the form: which may be rearranged to give the expression for energy: Note that the m0is the rest mass, and that m is the effective relativistic mass.
Energy a la Einstein Mass can be converted into energy with a yield governed by the Einstein relationship: where c = the speed of light. The yield from converting one kilogram is The energy consumption for one U.S. citizen for one year is about So one kilogram of mass conversion could supply the needs of about 180,000 U.S. citizens for one year, or the needs of a city of one million for over two months. * This amount will be used as a comparison unit when discussing energy production by nuclear fission and nuclear fusion
Some Nuclear Units A convenient energy unit, particularly for atomic and nuclear processes, is the energy given to an electron by accelerating it through 1 volt of electric potential difference. The work done on the charge is given by the charge times the voltage difference, which in this case is: Nuclear masses are measured in terms of atomic mass units with the carbon-12 nucleus defined as having a mass of exactly 12 amu. It is also common practice to quote the rest mass energy E=m0c2 as if it were the mass. The conversion to amu is:
Nuclear Binding Energy Nuclei are made up of protons and neutrons, but the mass of a nucleus is always less than the sum of the individual masses of the protons and neutrons which constitute it. The difference is a measure of the nuclear binding energy which holds the nucleus together. This binding energy can be calculated from the Einstein relationship: nuclear binding energy = Δmc2 For the alpha particle Δm= 0.0304 u which gives a binding energy of 28.3 MeV. The enormity of the nuclear binding energy can perhaps be better appreciated by comparing it to the binding energy of an electron in an atom. The comparison of the alpha particle binding energy with the binding energy of the electron in a hydrogen atom is shown below. The nuclear binding energies are on the order of a million times greater than the electron binding energies of atoms.
Fission and Fusion Yields Deuterium-tritium fusion and uranium-235 fission are compared in terms of energy yield. Both the single event energy and the energy per kilogram of fuel are compared. Then they are expressed in terms of a nominal per capita U.S. energy use: 5 x 1011 joules. This figure is dated and probably high, but it gives a basis for comparison. The values above are the total energy yield, not the energy delivered to a consumer
Fission and fusion can yield energy The binding energy curve is obtained by dividing the total nuclear binding energy by the number of nucleons. The fact that there is a peak in the binding energy curve in the region of stability near iron means that either the breakup of heavier nuclei (fission) or the combining of lighter nuclei (fusion) will yield nuclei which are more tightly bound (less mass per nucleon).
Proton-Proton Fusion This is the nuclear fusion process which fuels the Sun and other stars which have core temperatures less than 15 million Kelvin. A reaction cycle yields about 25 MeV of energy.
Nuclear Reactions in the p-p Chain This is the nuclear fusion process which fuels the sun and other stars which have core temperatures less than 15 million Kelvin. A reaction cycle yields about 25 MeV of energy. The modeling of these reactions is a part of the standard solar model. Note that both reactions which produce deuterium also produce a neutrino. Measuring the energy output of the sun and comparing it to this model allows us to predict the number of neutrinos that will hit the earth. The fact that early experiments detected only about a third of that number was called the "solar neutrino problem"
Even though a lot of energy is required to overcome the Coulomb barrier and initiate hydrogen fusion, the energy yields are enough to encourage continued research. Hydrogen fusion on the earth could make use of the reactions: These reactions are more promising than the proton-proton fusion of the stars for potential energy sources. Of these the deuterium-tritium fusion appears to be the most promising and has been the subject of most experiments. In a deuterium-deuterium reactor, another reaction could also occur, creating a deuterium cycle:
Pair Production Every known particle has an antiparticle; if they encounter one another, they will annihilate with the production of two gamma-rays. The quantum energies of the gamma rays is equal to the sum of the mass energies of the two particles (including their kinetic energies). It is also possible for a photon to give up its quantum energy to the formation of a particle-antiparticle pair in its interaction with matter. The rest mass energy of an electron is 0.511 MeV, so the threshold for electron-positron pair production is 1.02 MeV. For x-ray and gamma-ray energies well above 1 MeV, this pair production becomes one of the most important kinds of interactions with matter. At even higher energies, many types of particle-antiparticle pairs are produced.
Photon A photon moves with the speed of light in any frame; it cannot have a rest frame and its rest mass is zero, m0 =0. For a photon, the relativistic momentum expression = speed approaches zero over zero, so it can't be used directly to determine the momentum of a zero rest mass particle. But the general energy expression can be put in the form and by setting rest mass equal to zero and applying the Planck relationship, E = h f, we get the momentum expression = frequency
Space-Time and Four-vectors in Relativity In the literature of relativity, space-time coordinates and the energy/momentum of a particle are often expressed in four-vector form. They are defined so that the length of a four-vector is invariant under a coordinate transformation. This invariance is associated with physical ideas. The invariance of the space-time four-vector is associated with the fact that the speed of light is a constant. The invariance of the energy-momentum four-vector is associated with the fact that the rest mass of a particle is invariant under coordinate transformations. The space-time 4-vector is defined by The energy-momentum 4-vector is defined by is Lorentz-invariant is Lorentz-invariant Space-time of Special Relativity = Minkowski space
The scalar product of two space-time 4-vectors is defined by and the scalar product of two energy-momentum 4-vectors by Note that this differs from the ordinary scalar product of vectors because of the minus sign. That minus sign is necessary for the property of invariance of the length of the 4-vectors. (euclidean scalar product)
The length squared of the space-time 4-vector is given by The length of a 4-vector is invariant, being the same in every inertial frame. This invariance is associated with the constancy of the speed of light. This expression can be seen to be the equation of a sphere, with light propagating outward from the origin at speed c in all directions so that the radius of the sphere at time t is ct. The length squared of the energy-momentum 4-vector is given by The length of this 4-vector is the rest energy of the particle times the speed of light. The invariance is associated with the fact that the rest mass and the speed of light are the same in any inertial frame of reference.
The Light Cone c2t2 = x2 + y2 + z2 is the squared of the distance traveled by light in time t ct if we drop one of the space variable, for example z, we get the equation of a cone: (ct)2 = x2 + y2 future The Light Cone places an upper speed limit for all objects. Only "massless" particles can travel along the cone. For example, a photon ("a particle of light") is massless. Thus, our worldlines are confined to always be within the Light Cone. y past x only for events within the light cone the temporal sequence is fixed
Conceptual Framework: General Relativity Principle of Equivalence Experiments performed in a uniformly accelerating reference frame with acceleration a are indistinguishable from the same experiments performed in a non-accelerating reference frame which is situated in a gravitational field where the acceleration of gravity is g = -a. One way of stating this fundamental principle of general relativity is to say that gravitational mass is identical to inertial mass. One of the implications of the principle of equivalence is that since photons have momentum and therefore must be attributed an inertial mass, they must also have a gravitational mass. Thus photons should be deflected by gravity. They should also be impeded in their escape from a gravity field, leading to the gravitational red shift and the concept of a black hole.
a box = accelerated frame Inside the box a mass m feels an acceleration g = – a A a ray of light moving from a point A on the right wall, will reach the left wall at a lower point B, as the box accelerated up during the time the light went from A to B. Such a deflection is almost unnoticeable on Earth, due to the fast light speed B m g = – a No experiment can locally distinguish between a gravitational field and an accelerated frame Light must be deflected by gravity
Gravitational Deflection of Light Einstein's calculations in his newly developed general relativity indicated that the light from a star which just grazed the sun should be deflected by 1.75 seconds of arc. It was tested during the total eclipse of 1919 and during most of those which have ocurred since.
Gravity and the Photon The relativistic energy expression attributes a mass to any energetic particle, and for the photon The gravitational potential energy is then When the photon escapes the gravity field, it will have a different frequency Since it is reduced in frequency, this is called the gravitational red shift or the Einstein red shift.
Escape Energy for Photon If the gravitational potential energy of the photon is exactly equal to the photon energy then Note that this condition is independent of the frequency, and for a given mass M establishes a critical radius. Actually, Schwarzchilds's calculated gravitational radius differs from this result by a factor of 2 R = 9 Km for 3 solar masses R = 3 Km for Sun R = 9 mm for Earth’s mass
Gravitational Time Dilation A clock in a gravitational field runs more slowly according to the gravitational time dilation relationship from general relativity (this is distinct from the time dilation from relative motion) where T is the time interval measured by a clock far away from the mass. For a clock on the surface of the Earth, this expression becomes (g = GM / R2) This time dilation is about 1 part in 109: R=h in above expressions gives the time difference between two clocks at altitudes which differ by h
a box = accelerated frame Clocks A and Bemit, say, 10 signals per second. But receiver R moves up, and collects more signals, say 11. He then concludes that clock A has emitted 11 signals while clock B has emitted 10: clock B runs slower. A h = heightof box The difference in time intervals is due to the ratio between the speed reached by R during the signal transmission (v = g t = gh/c) and the speed of light: B R in agreement with previous slide. Equivalence principle: what happens in an accelerated frame must happen in a gravitational field
Hafele and Keating Experiment "During October, 1971, four cesium atomic beam clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip ... Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanosecond during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks." J.C. Hafele and R. E. Keating, Science 177, 166 (1972)
