Special Relativity

Special Relativiy

Special Relativity (Translated from Relativités et quanta clarifiés) Bernard Schaeffer PhD

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Special Relativiy

1. INTRODUCTION Special relativity originated one century ago from unsolved problems and various observations incompatible with the ideas of that epoch. Maxwell predicted the existence of radiation pressure, already imagined by Newton and observable with the Crookes radiometer. The Maxwell equations have been criticized because they were not conserved in the Galilean transformation. With the newtonian absolute movement, speed and time one predicted that light should be dragged by the Earth’s movement. Michelson-Morley experiment had to prove the existence of the Ether. The negative result of the experiment led to light speed invariance. Special relativity is special because it is limited to uniform translation, without any acceleration. Its fundamental postulate is the invariance of light speed in a change of Galilean reference frame. The Galilean transformation had to be replaced by the Lorentz transformation in order to take into account this experimental result, already known from the Maxwell equations. Einstein deduced directly the Lorentz transformation without using the Maxwell equations. From the Lorentz transformation one deduce various transformations : time, length, speed, acceleration, mass… Acceleration ought to be incompatible with Galilean frames but Einstein took the precaution of saying that special relativity should be applied to the "slowly accelerated electron". Using time as a fourth spatial dimension, one obtains the pseudo-euclidean space of Minkowski, euclidean by using an imaginary fourth dimension. Completed by Newton’s laws, special relativity became the relativistic dynamics whose principal application is the formula giving the energy contained in a mass at rest or in movement. The diagram below shows the logical process from the Lorentz transformation to E = mc2.

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Relativité restreinte Linéarité

c = cste

Equations de Maxwell

Réciprocité

Transformation de Lorentz Directe Réciproque

t =  t' + vx' c2 t' =  t - vx c2

Dilatation du temps: Immobilité de la règle dans le référentiel R' en mouvement x' = 0 t = t'

Vitesse limite = c

vx =

x =  x' + vt'

=

x' =  x - vt

2 1 - v2 c

Contraction des longueurs: Instantané depuis le référentiel R de l'observateur t=0 x = x' 

v' x + v v v' x 1+ c2

Accélération

Théorême de Pythagore s 2 = x 2 + y 2 + z 2 + ict 2 dans l'espace à quatre dimensions de Minkowski 2 ds 2 = dr 2 + r 2 d 2 + sin 2  d + d ict 2

Dynamique relativiste Loi de Newton F = d mv relativiste dt Masse relativiste m =  m 0 Energie cinétique E c = m - m0 c2 Energie proportionnelle à la masse

1

E = m c2

Flow chart of special relativity

d v = dv' dt dt'

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2. MICHELSON-MORLEY EXPERIMENT 2.1. THE AETHER The Michelson-Morley experiment consists to compare light speed in the directions parallel and perpendicular to the motion of the Earth on its axis. If Aether exists, still in an absolute reference frame, light speed should be constant in this reference frame, like sound speed in the air in the absence of wind. According to the Galilean principle of superposition, the speed of light is increased or decreased, depending on the direction and amplitude of the wind as may be shown with a ultrasound anemometer measuring the wind velocity based on the transit time of ultrasonic acoustic signals. There are two possiblilities, either the Aether is still relative to the Universe and the Earth is in motion, the speed of light will vary with the orientation, or the Aether is stuck to the Earth and light speed is independant of direction. Let us take a closer look to this experiment with the swimmer analogy.

2.2. THE MICHELSON SWIMMER Crossing a lake Let us consider a swimmer crossing a lake of width L0. The time t0 of a roundtrip crossing at speed c is given by 2L 0 t0 = c

Crossing a river To cross a river, the swimmer has to swim obliquely upstream in order not to be dragged downstream. The relative speed c of the swimmer has to be larger than the driving speed v of the current. If c = v, the swimmer stays on the spot and the duration of the crossing is infinite. If c < v his route seems oblique to an observer staying on a boat dragged by the current and perpendicular to an observer staying on the bank. In any case the duration of the crossing is larger with a current than without : this is a kind of time dilation ! Nevertheless, the time is absolute in classical mechanics and the swimmer has the same time as an observer on the bank or on a boat. In a given time, the distance covered by the swimmer is the vector sum of the distances covered along and across the river.

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The same is valid for the speeds obtained by dividing by the corresponding time increment. The absolute speed v1 is the swimmer speed for an observer on the bank. The absolute speed is perpendicular to the bank and given by the Pythagorean theorem : c 2 = v 21 + v 2 The duration of the round-trip crossing is : 2L 0 2L 0 t1 = = v1 c2 – v2 That is to say : t0 t1 = 2 1 – v2 c The crossing time increases with the speed of the current and becomes infinite when the speed of the current attains that of the swimmer in still water. Time seems to be dilated for an observer on the bank.

Swimming along the river The velocities add the way down and subtract the way up. The durations add. Therefore, the time t2 necessary for a round-trip along the river with the same distance L0 parallel to the current is : L0 L0 t2 = + c-v c+v that is t0 t2 = 2 1- v c2 This time dilation, with a slightly different formula, is larger.

Comparing the travel times For the same distance, it takes a longer time to swim along the river bed than to cross it. Both times are larger than in still water. The time difference between swimming perpendicular and parallel to the stream is :

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t2 – t1 =

t0 2 1– v c2

–

t0



v2 t0 2c 2

v2 c2 This is the formula that Michelson proposed himself to transpose to light. The swimmer velocity c is that of light (nowadays a photon). The current speed v is the velocity of the Aether wind. 1–

2.3. MICHELSON INTERFEROMETER The Michelson interferometer is a very sentitive equipment made of two perpendicular mirrors M1 et M2 and a half transparent mirror, inclined at an angle of 45 degrees, so that half the light pulse goes on through the glass, half is reflected. The two arms of the apparatus have equal lengths, are perpendicular and may rotate. One has two beams from the same light source, reflected parallel to the incident ray and coming again together through the semireflecting mirror. Equal optical paths may be adjusted very precisely in order to obtain interference fringes. The fringes should move with the orientation of the interferometer if the speed of light depends on that of the solar system (400 km/s). The shift should be maximum in the direction of the constellation Virgo. According to Michelson, the precision of the apparatus is even enough to detect the Earth’s Aether wind due to the rotation of the Earth around the Sun (30 km/s or 0.04 fringe). A later improvement with an eleven meters optical path, should even detect the Aether wind due to the rotation of the Earth on its axis (400 m/s or 0.005 fringe. One should note that there are already three absolute reference systems. The newtonian notion of absolute space is thus physically incorrect.

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The formula derived for the swimmer remains valid for the Aether wind : 2 t = t 2 – t 1  v 2 t 0 2c In order to minimize errors, the shift of the interference fringes, being measured at 0° and 90°, has to be multiplied by two : 2 2 v 2 Lv t  2 t 0 = c c3 The variation of the optical path is then : 2Lv 2 n  = c t  c2 The number n of fringes shifted by the translation at 30 km/s of the Earth around the Sun is, for a 600 nm wavelength, with c = 300.000 km/s and with arms of 1.2 m length 2 2 2  1.2  10 -4 2Lv n = = 0.04  c2 6  10 - 7 To their amazement, Michelson and Morley found that the velocity of light was independent of its direction of travel through space. There was no observable fringe shift although the expected effect was twice the experimental error. Michelson and Morley carried out, in 1887, a new experiment ten times more sensitive with the same null result : there is no Aether wind. The speed of light seems to be constant, even in single trip as shown by measuring the speed of light emitted by the -ions (or pions or pi-mesons). However, there still exist people who believe in the Aether, like the Nobe Prize winner Maurice Allais.

2.4. CONTRACTION AND DILATATION Going back to the formulae giving the crossing times : t0 t0 t1 = et t 2 = 2 v2 v 1 1– 2 c2 c t0 is the round-trip time in still water (v = 0) t1 is the round-trip time for crossing the river with a current of speed v t2 is the round-trip time along the river bed with a current of speed v The Michelson-Morley experiment having shown that, for light, these two times were equal. One has :

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2L 1

=

2L 2

2 v2 c 1– v c2 c2 L0 is replaced by L1, parallel to the movement, and L2, perpendicular to the movement. L1 and L2 are, indeed, the only adjustable parameters. In order to verify the preceding equality, one needs to have :

c 1–

2 v L2 = L1 1 c2 Fitzgerald had read a paper from Heavyside showing that the electric field of the moving charge distribution undergoes a distortion, with the longitudinal components of the field being affected by the motion but the transverse ones not. Then, if we assume that intermolecular forces are electrical, then we have L1 = L0 and

v2 c2 which is the Fitzgerald-Lorentz contraction, a consequence of the Maxwell equations. Then 2L 1 2L 0 t0 t1 = = = 2 2 2 c 1 – v2 c 1 – v2 1 – v2 c c c where t0 is the round-trip time in the absence of Aether. It shows that the time dilates. But the time is independant of the direction of movement. A clock, laser, for example will have a period increasing with speed, but without change under a slow rotation, even if its dimensions vary with speed. This should be true for any type of clock, mechanical, optical or electronical. Then t0 t1 = t2 = 2 v 1- 2 c The period of the pendulum of a clock A clock with one beat per second will have one beat in two seconds at a speed of 261,000 km/s for a fixed observer. L2 = L0 1 -

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3. RELATIVISTIC KINEMATICS 3.1. GALILEAN TRANSFORMATION According to newtonian mechanics, the the absolute velocity is the sum of the relative and the transferred (may also be called dragging or entrainment, in fluid mechanics) velocities : va = vt + vr For a constant speed, the abscissa is a linear function of the time. We may then write : xa = vt t + xr or with other notations : x = x' + v t' where t = t’ : the time does not depend on the reference frame. In fact, it does : time is different in New York and in Paris but the difference is a constant. We may also write it

x' = x  v t It is exactly the same, except for the sign of v with t = t’. It is the reciprocal Galilean transformation.

3.2. DERIVATION OF THE LORENTZ TRANSFORMATION The Galilean transformation needs to be generalized to take into account the constancy of the speed of light. The frame of reference R of the observer, generally considered as motionless, corresponds to the absolute reference frame of newtonian mechanics. The frame of reference R’ is the relative reference frame. The speed v is the classical transferred velocity, assumed to be constant. The Lorentz transformation, in its simplest form, is usually written in two dimensions (space and time), with the x axis coinciding with the velocity vector of R’ relatively to R.

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Linearity In special relativity t  t’. The simplest linear relationship between spacetime in the R and R’ reference frames is, with three independent coefficients, , , , function of the velocity v of the particle and v of the light is: x' =  x  vt

t' =  t + x This is the Lorentz transformation de Lorentz that becomes the Galilean transformation for  =  = 1 and  = 0.

Constancy of light speed In order to have a speed of light c independent of the reference frame, one needs to have x = ct and x' = ct'. The first of the preceeding equations then becomes : ct' =  ct - vt Using the second one t' =  t + ct we get :

c t + ct =  ct - vt Simplifying by t et dividing by c, one obtains the relation  1 + c = 1 v  c

Relativity principle According to the principle of relativity, there is no privileged reference frame. One has to find the same relationship when passing from R to R’ or, inversely from R’ to R. The relative speed v of the frames needs however a change of sign for the same reason as for the Galilean transformation. The direct transformation is ct' =  c  v t The reciprocal transformation of the abscissa is ct =  ct' + vt' then

ct' =  c  v t =  c  v or

 ct' + vt' c

Special Relativiy

c=  c  v

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 c+v c

After transformation this expression gives the Lorentz factor : 1 = v2 1 2 c The second Lorentz relation, using x = ct and  1 + c = 1 v  c becomes :

t' =  t + x =  1 + c t = 

 1 v  c

t=1 v t= c

t

vt c

2 v 1 c2

Replacing t = x/c, we get the Lorentz transformation of the time : vx t' =  t – 2 c

Algebraic form The constants  et  being determined, one obtains the direct Lorentz la transformation : t  xv x  vt c2 x' = t' = 2 v2 v 1 2 1 2 c c and the reciprocal t' + vx' 2 c x' + vt' t= x= 2 2 v 1 v 1 2 c2 c When light speed c tends to infinity, the Lorentz factor  tends towards one. The preceding formulae become the Galilean transformation : x' = x - vt et t' = t or x = x' + vt et t = t'

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where the resultant displacement is the sum of the relative and of the transferred displacements.

Matrix form The Lorentz transformation

x' =  x – vt t' =  t – xv c2 may be written in matrix form :

1

–v

x v 1 t c2 Using i =  1 , y = ict and y' = ict', we obtain : iv 1 c x x' y' =  iv y – 1 c Multipliantthe transformation matrix by its transpose, on obtains the unit matrix : x' t' = 

1 

– iv c

iv c 1

1 – iv c  iv 1 c

= 2

–

2 1 + iv c – iv + iv c c

2 1– v 0 – iv + iv c c c2 2 = 2 2 – iv + 1 0 1 – v2 c c

=

Its transposed is also its inverse. There is conservation of the lengths in the space x, y = ict which is then euclidian. The Lorentz transformation is then a rotation of an imaginary angle. In the littérature one fins a matrix presentation of the four-dimensional Lorentz transformation représented by the capital lambda (the L of Lorentz) : 1 – 0 0  ij =  –  1 0 0 0 0 1 0 0 0 0 1 where  = v/c. In an arbitrary direction, sans explicit it, on may write the general form of a linear transformation in the four-dimensional spacetime as :

1 0 0 1

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 ij

a11 a = 12 a13 a14

a12 a22 a23 a24

a13 a23 a33 a34

13

a14 a24 a34 a44

where the aij are of the form

aij =

x' i x j

= a i,j

where the partial derivative is indicated by a comma. The comma, representing a partial derivation, abridges considerably the formulae in relativity. In two dimensions the a linear transformation is : x' x' dx' = dx + dy = x' , x dx + x' , y dy x y y' y' dy' = dx + dy = y' , x dx + y' , y dy x y

Rotation in spacetime By putting ict = y, ict' = y', tg (i ) = iv/c, one has 1 1 1 = = = 2 2 1 + tg 2 i 1 + iv 1 - v2 c c

= cos i

The Lorentz factor  is real ; indeed

exp(i 2 ) + exp(– i 2 ) exp(– ) + exp() = = ch  2 2 The transformation becomes, in matrix form, a rotation of an imaginary angle i :  = cos i =

1 x' y' = 

– iv c

iv c 1

x y

= cos i

1 – tg i

tg i 1

x = y

cos i sin i – sin i cos i

Using the hyperbolic functions, one eliminates the imaginary quantities by replacing y = ct and y’ = ct' : x = x' ch  – y' sh  y = – x' sh  + y' ch  These are formulae analogous to those of rotation, where the trigonometric functions are replaced by hyperbolic functions. The terms in hyperbolic sines are preceded each with a minus sign. In the rotation, only one sine is preceded

x y

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with a minus sign. The Lorentz transformation matrix is symmetrical while the rotation matrix is antisymmetrical. The Lorentz transformation is then a hyperbolic rotation in a pseudo-euclidean space or in a true rotation in a euclidean space, but with an imaginary angle. In this euclidean space, the time is an imaginary distance, ict. The Lorentz transformation may be generalised in vectorial form for some rare practical applications.

3.3. TIME AND LENGTH Time dilation Let us consider a motionless observer in a reference frame R. He looks at a clock (not a pendulum clock, depending on the Earth gravity) moving at a velocity v. He measures a time interval t between two beats of this moving clock. An observer moving with the clock (x’ = 0) in R’ measures a time interval t’ between two beats of his clock. The second equation of the Lorentz transformation is : vx' t' + c2 t= v2 1 c2 With x’ = 0, we get

t=

t' 2

= t'

1 – v2 c The time interval between two beats looks larger for a moving clock. It becomes infinite when the speed approaches that of light. A photon is immortal. A meson has a limited life that can be measured practically motionless in the laboratory and at high speed in the atmosphere. A longer life was found at high speed than at rest in accord with the preceding formula. The twin paradox is something similar but usually misinterpreted. One compares two twins, one staying on Earth and the other flying with a rocket near the speed of light. The twin staying on earth will see the other aging slower. Now let us apply the principle of relativity : there is no preferred frame. Then the twin on the rocket will see the twin on earth also aging slower. Both of the twins will see the other one aging the same way, with or without acceleration and when they will meet again they will have the same age. Indeed, acceleration, being a

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differential of space to time, is relative between the twins like time, space and velocity. Within the scope of special relativity the acceleration is not absolute. It is often assumed without proof that there is a stationary and a travelling twin (relative to what absolute frame?). Time, space and their derivatives depend on the relative velocity between the frames. No reference frame is preferred.

Length contraction With the same kind of reasoning, let us consider an observer in a frame R measuring a length x of a ruler moving at a relative speed v in a reference frame R’. The observer in the moving frame R’measures a length x’. He has to take an instantaneous photograph, that is, t = 0. We use the first Lorentz equation x  vt x' = 2 1  v2 c where we put t = 0 :

x' =

x 2 1  v2 c

The length apparent to the motionless observer being x’, we have :

x = x' 1 

v2 c2

which is the Lorentz-Fitzgerald contraction. A direct measure does not seem to exist, but it is taken into account in the calculation of the synchrotron radiation, the diameter of the accelerator being different in the frame of the high speed electron and in the frame of the laboratory. Like the time, the lenth of a ruler parallel to the speed depends on the relative speeds of the ruler and the observer. A ruler contracts at high speeds while the time dilates.

3.4. COMPOSITION OF VELOCITIES In classical kinematics, velocities simply add vectorially according to the Galilean transformation.

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Colinear velocities In relativistic kinematics, near the speed of light things are more complicated. We shall limit ourselves first to a single spatial dimension, with colinear velocities. The Lorentz transformation is valid, in principle, only for Galilean reference frames, that is, for constant transferred speeds. The relative speed v between two Galilean frames R and R’ and the Lorentz factor  are constants. The Lorentz transformation may then be written in différential form : dx =  dx' + v dt' dt =  dt' + v dx' c2 with 1 = 2 1 - v2 c Using vx = dx/dt and v’x = dx’/dt’, we get the relativistic composition of velocities : dx' + v  dx' + v dt' v x = dx = = dt' dt 1 + v2dx'  dt' + v dx' c dt' c2 that is :

v' x + v v v' 1+ 2x c For an infinite light speed, the denominator is equal to one. We then recover the classical formula of speed addition where the absolute velocity vx = va is the sum ot the transferred velocity v = vt and of the relative velocity v’x = vr : va = vt + vr In einsteinian relativity, velocities add as in Galilean relativity except that a factor prevents to reach the speed of light. Let us chek it. If v’x = c, as for a photon in a frame moving at speed v, then we have : vx = c + v = c c + v = c vc c+v 1+ 2 c The velocity of a photon does not depend on the speed of the reference frame. vx =

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The relativistic composition of the velocities is no more the Galilean addition of velocities. There is a factor preventing to overpass light speed. For low velocities one recover the classical principle of superposition of velocities.

Non colinear velocities Let us consider two frames R and R’ whose axes Ox and O’x’ coincide, their origins O and O’ moving away from each other with velocity v. Considérons des référentiels R et R’ dont les axes Ox et O’x’ coïncident, les origines O et O’ s’éloignant l’une de l’autre à la vitesse v. Les composantes des vitesses seront donc The components of the velocity are vx and vy in R, v’x and v’y in R’. The relation between vx and v’x is the same as for colinear velocities. Using the differential form of the Lorentz transformation, one has dy = dy’, in the absence of transverse contraction : dt =  dt' + v dx' =  1 + v2 dx' dt' 2 c c dt' which gives dy' dy' dt' v y = dy = = dt  dt' + v dx'  1 + v dx' 2 c c 2 dt' and lastly

v' y

1

vy = 1+

v2 c2

vv' x

c2 This formul is used to calculate the relativistic aberration.

3.5. LONGITUDINAL DOPPLER The Doppler effect is observed when a vibrating source whose frame is R’ emitting sound or light waves approaches or moves away from the observer, as for example, a noisy motorcycle. When the source is moving towards the observer whose frame is R, the center of each new wavefront is slightly displaced towards him. The wavefronts begin to bunch up towards the observer and spread further apart behind the source. An observer in front of the source will hear a higher frequency, and an observer behind the source will hear a lower frequency.

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The same happens for electromagnetic waves from radars or lasers that are used to measure speeds. Il is also observed for redshifted spectral lines emitted by galaxies at the origin of the expanding universe and Big-Bang theories. There is also a Doppler effect due to matter emission in supernovae. A similar effect is the redshift due to gravitation at the surface of the stars that may be considered as a Doppler effect only through the principle of equivalence of general relativity. In classical physics, the relative frequency shift, as seen by the observer is :  – s  v = r =– s s c where vr is the velocity of the receptor and vs the velocity of the source. v is the relative velocity between the source and the receptor, positive when the observer (the receptor) goes away from the source. The frequency decreases when v > 0. This formula needs only to be multiplied by the Lorentz factor  to remain valid when v approaches the speed of light as we will show. The Lorentz transformation of the time : vx' t' + 2 c t= 2 v 1c2 becomes, for a light ray of velocity c, with x’ = ct’, t =  t' + v c2 t' =  1 + v t' c c For one période, that is t = T in R and t’ = T’ in R’ : 1+ v c T' T =  1 + v T' = c v 1– c The frequency being the inverse of the period, one has : v 1–  c = ' 1+ v c When the velocity is positive, that is when the source and the receptor move away from each other, the frequency perceived by the observer is lower. At low speeds,   1, we may develop the formula up to the second order : r – s v – vs v v2 1 vr – vs 2 – + =– r + s c c 2 c 2 c2 It differs from the classical formula by the second order term :

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r – s vr – vs v 2r – + 2 s c c For velocities near the speed of light, with a negative velocity, v   c, we obtain the ultra-relativistic formula : r = 1– v  1+ c =2 s c c The frequency increases indefinitely with the velocity. This formula is used in the theory of the synchrotron. With a positive velocity, the frequency tends to zero.

3.6. RELATIVISTIC STELLAR ABERRATION The stellar aberration is similar to rain falling along the window of a train. The rain is falling vertically when the train is at rest and inclined when the train is moving. When the speed of the train is much larger than the velocity of the falling rain, the rain appears to move horizontally. Let us make a a simple thought experiment with a vertical tube standing up under the rain. The rain falls in it to its bottom without touching the inner wall of the tube. Now let us move : the rain will no more attain directly the bottom of the tube. In order to do it, we have to incline the tube from an angle  such that tg  = v/c, ratio of the velocity v of the falling rain and your speed c. This formula, purely geometrical, has nothing to do with relativity. By replacing rain by light from a star at infinity, one may do the same experiment with a telescope. With v = 30 km/s, the velocity of the Earth around the sun and c = 300.000 km/s, that of light, the angle is  = 21" = 10-4 radian, for the annual aberration of stars. This calculated value is in accord with the numerous observations made since the 18th century by Bradley. In order to show that the phenomenon does not occur inside the telescope, Airy showed, by filling the telescope with water, that the refractive index had no influence. The stellar aberration should not be confused neither with the optical aberration of optical instruments nor with the parallax of stars near the Earth. The stellar aberration is a phenomenon similar to the Doppler effect but concerns the direction of propagation instead of its frequency. Let us now calculate the relativistic aberration. One may consider, according to the principle of relativity that the star moves along the x’ axis of the reference frame R’ at the velocity v of the Earth, motionless relative to the terrestrial observer in the frame R. The axis x and x’ coincide. The light ray, with velocity c, is inclined at an angle relative to x. The projections on the axis x’ and y’ are x’ = ct’ cos ’ and y’ = ct’ sin ’

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The Lorentz transformation equations write, with  = v/c : x =  x' + vt' =  ct' cos  +  y = y' = ct' sin  By making the ratio y/x, one eliminates t’ to obtain the relativistic formula of the aberration of light : sin ' tg  = y = x  cos ' +  For a star at the zénith, the angle is ’  90°, that gives 2 tg  = 1 = c 1 – v  v c2 When the velocity is low, the angle with the vertical line being also low,  = ’ – is near 1/tg and one finds again the classical aberration of stars : v   c An electron moving at a velocity low relatively to that of light emits electromagnetic radiation in a wide range of directions. At a relativistic velocity, near the speed of light, the light emission is concentrated towards the front of the electron, the angle tends to zero : v2  1– 2 c The stellar aberration should prove that the Earth moves relatively to a referential frame bound to the Aether (invented by Maxwell !). The Michelson experiment had shown that the velocity of light was not influenced by that of the Earth. To explain this, it was imagined that the Earth’s gravitational field somehow “dragged” the aether around with it in such a way as locally to eliminate its effect. If the velocity of the Aether were local to the Earth, the stellar aberration would vary with the altitude, which is not the case. Therefore stellar aberation is incompatible with the absence of Aether wind. The stellar aberration is explained geometrically in classical kinematics. For relative velocities approaching the speed of light, a relativistic correction is needed. For example, the synchrotron radiation is concentrated towards the front of the electron beam.

3.8. TRANSFORMATION OF ACCELERATIONS Changement of reference frames is more complicated for accelerations than for velocities. We shall restrict ourselves to rectilinear motion and to uniform circular motion.

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Classical kinematics In classical rational mechanics, the term "Galilean transformation" was not in use. One said only that kinematics differed from geometry by the introduction of time. It seemed natural that displacements add vectorially. Velocity was simply the vector derivative of displacement and acceleration the vector derivative of velocity. For the sake of simplicity let us stay in only one space coordinate. Let us consider the acceleration of an electron in a electric field. Let R be the reference frame of the laboratory and R’ a Galilean reference frame. Let x and x’, vx and ax, v’x and a’x, respectively, abscissas, velocities and accelerations of the electron in frames R and R’. According to the Galilean transformation, the velocity is the derivative of the abscissa. For a constant velocity, we have x = x’ + vt By dérivation, we get dx/dt = dx’/dt + v or v’x = v’x + v This formula remains valid even if v varies. After a subsequent derivation we obtain the acceleration : 2 2 dv = a' + dv a x = d 2x = d dx' + v = d x' + x dt dt dt dt dt dt 2 If the frame R’ coincides with the electron v’x = 0 and a’x = 0. Then : dv ax = dt We don’t need Galilean reference frames to know the acceleration. Let us see what happens when using the Lorentz transformation.

Acceleration parallel to velocity Simple me thod In relativity, when changing from a frame R to a frame R’ of relative velocity v, time dilates with speed and length parallel to the velocity contracts according to the formulae : dt =  dt' et dx = dx'/ Let vx and ax, v’x and a’x, respectively, velocities and accelerations of a particle with abscissas x et x’ in frames R and R’ of the motionless and mobile observers. The acceleration is the second derivative of space relative to time.

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Therefore, length being divided by  and time multiplied by , the acceleration has to be multiplied by 3. The acceleration is then : d 2 x' 3 2 2 2  d x v – 3 d x' ax = 2 = = = 1  2 2 a' x 2 2 d t' dt dt' c The acceleration is smaller for the observer than for the particle. We may also write : dv dv' x 3 x = dt dt' If the electron is motionless in frame R’, then vx = v. In relativistic kinematics, we need to take into account the Lorentz factor (v) with v variable. Let us compute

3 =

1 2 1 – v2 c

3

= d dv

v 2 1 – v2 c

=

d v dv

which is a formula found by Lorentz in his "Theory of the electron". Using this result, we may rewrite the proper acceleration : d v dv' = dt' dt In the frame of the electron, the proper velocity is v’ = 0 but the proper acceleration dv’/dt’ is not zero. This true also in classical mechanics. A passenger at rest in the frame R’ of a lift will feel an acceleration with no motion. In a relativistic speed, the acceleration measured in the lift, with an accelerometer, is different from the acceleration measured from the ground, with an optical method. The proper acceleration of a particle is simply obtained by deriving, relatively to the apparent time t, the apparent velocity v multiplied by the Lorentz factor . The relativistic acceleration measured by the observer differs from the proper acceleration while they are equal in classical mechanics. The acceleration is zero at the speed of light. Bette r meth od We have seen that the Lorentz transformation of the velovity is: v' x + v vx = v v' 1+ 2x c The acceleration being the derivative of the velocity, we have, with v = constant for Galilean reference frames R and R':

Special Relativiy

dv x = dt

23

v' x + v v' x + v dv' x 1 = v v' x v v' x dt v v' x 1+ 1+ 1 + c2 c2 c2

v dv' x c 2 dt

2

Then, after simplification:

v2 1- 2 dv x c = dt v v' x 1+ c2

2

dv' x dt

The Lorentz transformation of the time increment is:

v v' x v dx' v dx' 1 + 1 + c2 = c 2 dt' dt' = c 2 dt' 2 2 2 1- v 1- v 1- v c2 c2 c2

dt' + dt =

Remplacing dt on the right of the above expression of the acceleration we obtain the acceleration ax in the observer's frame as a function of the proper acceleration a'x:

v2 1dv x c2 ax = = dt v v' x 1+ c2

3

dv' x = dt'

v2 1c2 v v' x 1+ c2

3

a' x

We may now put v'x = 0. If we had done it before derivation,we would have gotten a null acceleration. The same thing would happen in classical kinematics. We have then the above formula: 2 dv x v ax = = 1- 2 dt c

3 2

dv' x =  - 3 a' x dt'

Constant proper acceleration An example of a constant proper acceleration g may be an electron accelerated in a constant electric field or a mass in the constant gravity near the surface of the Earth : dv' x a' x = =g dt' We have then a differential equation :

24

Special Relativiy

d v =g dt or

d v = d gt that integrates into v = gt. The integration constant is zero if the initial velocity is v = 0 at t = 0. With 1  = v 2 1 c One may write after integration :

v =

c

2 1+ c gt For slow speeds, that is for c =  and t = 0, the formula becomes v = gt. When t increases indefinitely, the velocity approaches asymptotically the speed of light. The apparent acceleration, for the observer in the R frame, decreases continually toward zero but remains constant in the mobile reference frame R’. This formula, used in particle accelerators, may be written : d 1 2 gt dt 2 dx v= = 2 dt 1 + gt c When t is small, the denominator is equal to one, giving the classical law of falling bodies. When time increases, the velocity continues to increase, but at a decreasing rate. The infinitesimal displacement gt 2 d 2 c dx = c 2 2g 1 + gt c may be integrated as 2 2 c x= 1 + gt g c After some algebra, one gets the equation of a hyperbola : 2 x2  c2 t 2 = c 2 g This is the reason why the relativistic uniformly accelerated movement is called hyperbolic.

Special Relativiy

25

Variable proper acceleration We had obtained above the formula giving the relativistic acceleration:

d v =g dt This formula remains valid for a variable acceleration like gravitation:

d v GM d GM = 2 = dt dr r r For a radial velocity, we may write v=dr/dt, which gives

d v d GM = dr dr r

v and thus:

vd v = d GM r Now, we have the identity

d = d

1 1–

v2 c2

1 v2 = 1– 2 2 c



3 2

3 2v v  2 dv = d 2 c c

2

=

v d v c2

which gives

d = d GM c2 r and integrates in

1

= GM + constant c2 r

2 v 1– 2 c

The gravitational potential energy of a proper mass m0 is: GM V =  m0 r Multiplying both sides by the proper mass m0 of the particle and by c2, we obtain the relativistic conservation of energy:

1

m 0 c2

2

1– This is indeed T + V = constant.

v c2

 1 + V = constant

Special Relativiy

26

v=

dx =c 1– dt

1 1+

V m 0 c2

2

By using the lorentz transformation of the acceleration and assuming that the acceleration derives from a potential, we have obtained the relativistic conservation of energy. From it, we deduced the relativistic velocity of a particle in function of the potential. This approach is not valid for a photon in a gravitational field.

Acceleration perpendicular to velocity Acceleration is the second time derivative of the abscissa y, dy/dt, now perpendicular to the velocity dx/dt. According to the Lorentz transformation, there is no transverse contraction ; then y is not affected by the frame change : y = y’. Only the time is dilatated. We have : 2 d2 y d 2 y' v2 – 2 d y' –2 ay = 2 = = =  a' y = 1  2 a' y d t' 2 dt dt' 2 c This formula may be applied to electrons accelerated in a synchrotron where the speed is practically v  c. The acceleration is centripetal and perpendicular to the velocity, the trajectory being circular with radius r. The acceleration a’y in the frame R’ of the electron determines the radiation : 2 2 2 c a' y =  a y =  r where r is the bending radius of the synchrotron as seen in the frame R’ of the electron. c is the velocity of the electron, almost equal to the speed of light, equal in R and R’. The classical Larmor formula gives the radiation power emitted by the electron : 1 2e 2 a 2 P = 4  0 3c 3

By replacing the acceleration in the frame of the electron we get : 2 2 2 q 2 c 4 q 2 c P =  = r 6  c 3 6  r 2 0

0

It is also important to know the frequency of the radiation. At low speeds, the frequency is the Larmor frequency L, obtained by equating the centrifugal force and the Lorentz force m 0 v  L = evB in SI units as everywhere in this book. The Larmor frequency is also L = v/r :

Special Relativiy

L =

27

evB eB v = = m0 v m0 r

It is no more necessary to know the magnetic induction, replaced by the radius of the synchrotron, much easier to grasp. At the speed of light, v = c and the radius r is contracted according to the Lorentz factor. The frequency of the fundamental mode is then : c 0 =  r There is both a relativistic Doppler and a relativistic aberration. Both multiplie the frequency by . The so-called critical frequency of the synchrotron is then : c C  3 r The spectrum produced by the synchrotron extends in a practically continuous manner from the fundamental frequency 0 to the critical frequency c. The use of the special relativity theory avoids the use of retarded potentials and simplifies greatly the calculation of the Larmor formula at relativistic speeds.

3.9. DIFFERENTIAL OPERATORS Wave propagation is obtained by solving partial differential equations where differential operators appear.  instead d are used in the presence of more than one independant variable. We shall see how these total and partial derivative operators 2 d   2 , , , et dt x t x 2 t 2 transform in the Lorentz transformation. The total differential of a function f(x, t)is the same in the "motionless" frame R and in the "mobile" frame R’ : f f f f df = dx + dt = dx' + dt' x t x' t' Let us express the partial derivatives in R’ with the help of the Lorentz transformation in differential form : dx' =  dx – v dt dt' =  dt – v dx c2 Replacing dx’ and dt’ in the above total differential of f in R’ : f f df =  dx – v dt +  dt – v dx x' c 2 t' By grouping the terms in dx and dt one gets :

Special Relativiy

28

f f v f f – dx +  – v dt x' t' c 2 t' x' By equaling both expressions of df, we have : f f f f v f f dx + dt =  – dx +  – v dt x t x' t' c 2 t' x' Identifying the dx and dt terms and suppressing the f we obtain the partial derivative operators :    = – v2 x x' c t'    = –v t t' x' df = 

It is to be pointed out that the minus signs are here on the side of the primed variables. They are on the unprimed side in the original Lorentz transformation above. We have also the reciprocal expressions:

 =  + v  x' x c 2 t

 =  +v  t' t x These formulas will be used to show the invariance of the electromagnetic wave equation in the Lorentz transformation. When c = ,  = 1, one obtains the Galilean transformation of the operators. The particle being motionless in its proper reference frame R' with velocity v relative to R, we have dx’/dt’ = 0. The derivative of f with respect to t’ is then: df f dx' f f = + = dt' t' dt' x' t' Using the preceding expression of /t’ one obtains: df =  f + v f =  df dt' t x dt Putting  = 1, on gets the formula of the material derivative of newtonian fluid mechanics. Using the proper time = t', we may write the total derivative operator: d = d d dt The partial derivatives are different in both classical and relativistic kinematics. The total derivatives are equal in classical kinematics but different in relativistic kinematics. Anyway, putting  = 1 gives always the classical formula to which one may refer in case of doubt about signs. In case of doubt about the position of , it suffices to remind that the classical formulas are valid in the proper frame.

Special Relativiy

29

These formulas will be used to check the conservation of the wave equations in the Galilean and Lorentz transformations.

3.10. WAVE EQUATIONS d’Alembert equation From the Maxwell equations one may obtain the d'Alembert equation where the celerity is that of light:

2  2

2 

+

2

+

2  2

1 2 

–

2

2

=0

x y z c t where  is the function representing the amplitude of the wave in the frame R. We shall show that this equation is conserved in a Lorentz transformation. The d’Alembertian operator 2 2

+

2 2

+

2 2



1 2 2

2

=+

2 2

x y z c t  ict may be written, for the sake of simplification, in a two dimensional spacetime R, x for space and t for time: 2     1 2 – = + – x c t x c t x 2 c 2 t 2 We have seen that, in the Lorentz transformation, the differential operators transform as:    = – v2 x x' c t'    = –v t t' x' where v is the velocity of R’ relative to R and 1 = 2 1 – v2 c is the Lorentz factor. Replacing these operators by their expression in the wave equation, one obtains:     + =  1 –- v + x c t c x' c t' and also, by changing c in – c :

Special Relativiy

30

   v  – = 1+ – x c t c x' c t' which gives the wave equation:    1– v + c x' c t'

 1+ v c

  – x' c t'

=0

or 2     2 1 – v + – =0 c t' x' c t' c 2 x' which is the original equation since v2 2 1 – 2 = 1 c The wave equation is the same in R and R'. The variables are primed in R' and unprimed in R. The wave function  and its celerity c are unchanged. The electromagnetic wave equation is invariant in a Lorentz transformation but the celerity has to be that of light in the vacuum. A sound wave equation has the same form in the absence of entrainment but is not invariant under a Lorentz transformation.

Hertz equation The so-called Hertz equation is the equation of mechanical waves, valid with entrainment, for example in a wind of velocity v:

2 

2

1 d  =0 2 2 2 c dt x  is the wave function that may be the density, pressure, stress, strain, volume, displacement… The main difference with the d’Alembert equation is the presence of straight d's for a total derivative operator instead of round 's for a partial derivative operator in the time derivative. The celerity is not the celerity of light but that of mechanical waves. One may often find this equation with round 's in the literature but it is correct only in the absence of entrainment. We shall check that it is invariant in the Galilean transformation. We will write explicitely the convective term:

2 x

2



1 2

–

  +v t x

2

=0

c where v is the entrainment velocity. In a Galilean transformation with velocity u the "absolute" velocity in frame R becomes v = v’ + u where v' is the velocity in

Special Relativiy

31

the moving frame R' and u the velocity of R' relative to R. The derivation operators may be obtained by putting c = ,  = 1 in the Lorentz transformation:   = x x'    = u t t' x' Using v = v’ + u and these expressions, the total derivative operator becomes:          +v = –u +v = + v–u = + v' t x t' x x t' x' t' x' The velocity u of R' is eliminated. The total derivative operator is the same in the R' frame except for the primes. The Hertz equation of waves, also called non-linear, is therefore conserved in the Galilean transformation:  2  2 1  – 2 + v' =0 x' x' c t' It may also be deduced, without calculation that it is also conserved in the Lorentz transformation. This is true only if the velocity of light is used in the Lorentz transformation and the celerity of the mechanical waves in the Hertz equation. The mechanical wave equation is not conserved in a pseudo-Lorentz transformation where the same c is used in both the wave equation and the Lorentz transformation. A d'Alembert equation will not work in the wind.

3.11. MINKOWSKI SPACE-TIME Minkowski metric The Pythagorean theorem is conserved in a rotation since lengths are conserved. We have seen that the Lorentz transformation is equivalent to a rotation of an imaginary angle i such that tg (i ) = iv/c : cos i sin i x x' y' = y – sin i cos i The length s of a segment has to be conserved in a rotation,in vertue of the Pythagorean theorem : s 2 = x' 2 + y' 2 = x 2 + y 2 which gives, when y et y’ are replaced by ict and ict’ : s 2 = x' 2 + ict' 2 = x 2 + ict 2 = x' 2 - c 2 t' 2 = x 2 - c 2 t 2 A minus sign, due to the square of i appears. The euclidean planar space is transformed in a flat pseudo-euclidean space called Minkowski space-time. It

Special Relativiy

32

may directly checked, by replacing t' and x' by their expressions issued from the Lorentz transformation: x' =  x – vt t' =  t – vx2 c that the métric 2 t  vx2 x  vt 2 c s' 2 =  c 2 t' 2 + x' 2 =  c 2 + 2 2 v v 1 2 1 2 c c is conserved after developing and simplifying:

s' 2 =

1

2 x2 1  v + v2  c2 t 2 c2

=  c 2 t 2 + x 2 = s2

v2 c2 The Minkowski metric is conserved by a Lorentz transformation and is easier to use than the Lorentz transformation. In general relativity, there is no practical transformation. 1

Cartesian coordinates The three-dimensional physical space is no more absolute since Einstein. Minkowski has shown that the phenomena discovered by Lorentz and clarified by Einstein could be described with a four-dimensional space. If the fourth dimension is defined as the distance travelled during the time t multiplied by i where i the square root of – 1 one obtains a four-dimensional euclidean space: s2 = x 2 + y 2 + z 2 + w 2 where

w =  c 2 t 2 = ct  1 = ict Without i, it is the Minkowski pseudo-euclidean space. The proper distance is then given by the metric: s2 = x 2 + y 2 + z 2  c 2 t 2 It is recognized by the minus sign before the t2 term. Some authors put c = 1. The Minkowski space-time is a euclidean space deformed by the combination of uniform dilatation and shear. It is therefore without curvature and hence a "flat" space with constant coefficients of the metric. It is not euclidean since the coefficients are not equal to one. In the ordinary euclidean space, lengths are conserved by translation or rotation. In the Minkowski space-time, the rotation is replaced by the Lorentz transformation.

Special Relativiy

33

We shall now define more precisely the notion of metric. In euclidean threedimensional analytic geometry, the spatial distance dl between two near points is:

dl 2 = dx

2

+ dy

2

+ dz 2 =

dx i dx i = dx i dx i i

One uses the differential notation although it is not necessary in special relativity where the movements are uniform, without acceleration. The sign  may suppressed thanks to the Einstein convention where repeated indices denote summation (not always) over their range. The generalized distance (or spacetime interval) ds between two events becomes in the Minkowski space-time: ds 2 =  c 2 dt 2 + dx 2 + dy 2 + dz 2 = – c 2 dt 2 + dl 2 dl is the displacement in the physical space during the time dt at velocity v. There are no more parentheses here. This simplified writing is not really correct but it is commonly in use. Indeed dx2 = 2x dx  (dx)2. Some authers use parentheses but their formulas are not very readable. The metric may also be written as: dl 2 2 v2 ds 2 =  c 2 dt 2 + dt =  c 2 dt 2 + v 2 dt 2 =  1  2 c 2 dt 2 < 0 dt c In relativity, the velocity v being less than the speed of light c, ds2 is negative therefore ds is an imaginary number. For that reason, one prefer often to use the proper time : 2 2 2 2 d 2 = – ds2 = dt 2 – dx + dy2 + dz > 0 c c Using the physical velocity

v=

dx 2 dy 2 dz 2 + + dt 2 dt 2 dt 2

the metric writes

v2 dt 2 > 0 2 c and is real. The velocity v of a photon is equal to the speed of light c, then d = ds = 0. The trajectory of a photon is a staight line The length of the trajectory is zero since all ist elements have a zero length. A massive particle has always a velocity less than the speed of light. When its velocity is zero, that is when the particle is motionless in its proper frame, we have d = dt. At low velocities, d  dt with d < dt. The proper time is always smaller than the physical time. In four-dimensional Riemannian geometry, the metric is generalized as follows: ds 2 = g ww dw 2 + g x x dx 2 + g y y dy 2 + g zz dz 2 = g i j dx i dx j d 2 = 1 –

34

Special Relativiy

where w = ict. The gij are called coefficients ou components of the metric. A four-dimensional metric tensor may be represented by a matrix: g ww g wx g wy g wz g00 g01 g02 g03 g wx g x x g x y g xz g g g g ou 0 1 1 1 1 2 1 3 g wy g x y g y y g yz g02 g12 g22 g23 g wz g xz g yz g zz g03 g13 g23 g33 Generally the indexes w, 4 or 0 correspond to the time. The matrix is symmetric, in the diagonal terms like gxy dx dy, dx and dy may be commuted without changing the value of ds2. Therefore gxy = gyx. Practically, for the sake of simplicity, we shall use almost always diagonal matrices, without gxy, gxt… as in the Minkowski metric: ds 2 =  c 2 dt 2 + dx 2 + dy 2 + dz 2 = d ict 2 + dx 2 + dy 2 + dz 2 that may be written, as in a Riemannian space: ds 2 = g t t d ict 2 + g x x dx 2 + g y y dy 2 + g zz dz 2 where gtt = gxx = gyy = gzz = 1. gtt, gxx, gyy and gzz are the only non-zero components of the metric Minkowski tensor. They are all equal to one, the minus sign appearing only when the square of i is carried out. The signs of the coefficients may vary according the conventions used. The sign of gtt is usually opposed to the others but it seems preferable to use (ict)2 instead of  c2t2 or even ± t2 with c = 1 which forbiddens any checking with dimensional analysis. The Minkowski metric is represented by a 4  4 diagonal matrix : -1 0 0 0 g i j = 0 1 0 0 = i j 0 0 1 0 0 0 0 1 or 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 if the metric of type ds2 or d 2 (d 2 is sometimes called ds2, in a so-called WestCoast or Lorentz metric). The ij désignate the gij of the Minkowski metric. All the diagonal ij are equal to ± 1, using a physical or geometrical unit system. The determinant is g =  1 or g = 1 if the fourth dimension is ict. In this latter case, the diagonal terms are all equal to one. In general relativity, the coefficients of the metric are function of the coordinates trough the gravitational potential and the Minkowski space becomes tangent to the curved pseudo-Riemannian space-time. We shall consider the space-time of general relativity as a four-dimensional Riemannian (not pseudo-Riemannian)

Special Relativiy

35

space with w = ict (Einstein uses x0 = ct) in order to avoid the minus sign problem.

Spherical coordinates Spherical coordinates are defined as the position vector r, the colatitude and the longitude  :

Let us consider the small spherical rectangle on the sphere. Its width is r sin d and its height is r d . The Pythagorean theorem may be applied to this rectangle to obtain its diagonal:

r d = r d 2 + sin  d

2

Simplifying by r, d gives the metric on the sphere. We may similarly increment r with dr to obtain a new rectangular triangle

dl = dr 2 + r 2 d 2

Special Relativiy

36

A last step gives the length element in a four-dimensional euclidean space with the fourth dimension w = ict:

ds = d ict 2 + dl 2 = d ict Replacing d we obtain the full metric:

2

+ dr 2 + r 2 d 2

ds 2 = d ict 2 + dr 2 + r 2 d 2 + sin 2  d 2 which is the pseudo-euclidean Minkowski metric: ds 2 = – c 2 dt 2 + dr 2 + r 2 d 2 + sin 2  d

2

or, in matrix form:

gtt 0 gij = 0 0

0 0 0 g rr 0 0 0 g  0 0 0 g

±1 0 = 0 0

0 1 0 0

0 0 r2 0 r2

0 0 0 sin 2 

In radial symmetry the metric simplifies: ds 2 =  c 2 dt 2 + dr 2

3.12. RELATIVISTIC LAGRANGIANS Variational calculus The calculus of variations is issued from the principles expressed by Heron of Alexandria, Huygens, Fermat, Hamilton, d’Alembert, Maupertuis and also from the works of Lagrange, Euler and others. The Lagrange equations may be obtained either from variational principles or Newton's laws. These ideas may be resumed by the principles of the shortest way (geometric aspect) or of the least effort (mechanical aspect). The effective trajectory is the one corresponding to the extremal way or time. The derivative of the way has to be zero all along the way.

The shortest way in a plane In order to find the shortest way from one point to the other, for example on a surface, one has to know the metric giving the shortest distance between two nearby points. It is important to define the metric in terms of differential changes in the coordinates since not all coordinate systems are linear like the Euclidean ones. The Pythagorean theorem defines the metric of the plane where the line element is given by: ds 2 = dx 2 + dy 2

Special Relativiy

37

On a surface, the Pythagorean theorem is generalized by the formula invented by Gauss:

ds 2 = g x x dx 2 + 2 g x y dx dy + g y y dy 2 The shortest way between two points A and B in the plane is:

S=

B

B

B

dx 2 + dy 2 =

ds = A

A

B

B 2

1 + y dx = A

L dy dx dx

L y dx = A A

where the Lagrangian is

L y = 1+ y

2

and

y = y' = dy dx is the slope of the curved way. The symbol  (lower case delta) instead of d (straight d for total differential), or  (curly d for partial derivative), shows a virtual infinitesimal variation. Developing the Lagrangian L in the first order, we may write the virtual variation of the way ds:

 ds = L y dx =

 L y d L y  y dx = y dx y  y dx

The differentiation being commutative, one may write: L y L y  ds =  dy dx =  y dx dx y y Expliciting the Lagrangian, we get:   ds = 1 + y x 2  y dx y Let us write:  A= 1 + y 2 et B = y y Let us integrate by parts this expression : 2

1

 y

2

1+ y

2

d y = dx

2

A dB = dA B dx dx 1

2

2

B dA = 0 – dx

–

1 1

The integrant has to be null whatever y: d  1 + y2 = 0 dx  y

1

 y d dx  y

1 + y2

38

Special Relativiy

or, using L:

d L = 0 dx  y Carrying the partial derivative relative to y', the Lagrange equation becomes: 2y x d =0 dx 2 1 + y x 2 It integrates in

y x 2

= constant

1+ y x or y’ = dy/dx = constant. The trajectory y(x) is a straight line.

Special Relativiy

39

4. RELATIVISTIC DYNAMICS 4.1. INTRODUCTION The Lorentz transformation, like the Galilean transformation is supposed to be valid only between Galilean reference frames. We have seen above that the observed acceleration is a function of the proper acceleration through the Lorentz formula of the accelerated electron. Accelerated motion is therefore not out of the scope of special relativity. Relativistic dynamics is the special relativity with addition of Newton' laws. Mass dilatation results simply from the application of the Lorentz transformation to the acceleration as we shall see. If the acceleration is defined as the derivative of the velocity, Newton's second law must be written with the variable mass included in the derivand:

F=

d mv dt

Energy being the product of the force and the displacement, as in classical mechanics, one obtains an expression that reduces to the newtonian formula at low speeds. When the effort F acts on a body and make it move of an increment dx, the work done by F is transformed into kinetic energy dT = F dx. By integration of this equation, one obtains the kinetic energy  mv2 in newtonian mechanics where the mass is constant. In relativistic dynamics, we have to take account of the variable mass, function of the velocity. We will show that the kinetic energy depends only on the mass via the velocity and a universal constant proportionality factor.

4.2. RELATIVISTIC MASS The Lorentz transformation of colinear accelerations is given by the formula: d v = dv dt d where t is the time in the frame R of the observer ; t’ is the proper time in frame R’ of the particle ; v is the velocity of frame R’ relative to R as defined for the Lorentz transformation. The accelerations are the same in both frames when  = 1 e.g. for velocities low relative to the speed of light. Let us multiply both sides of the preceding equation by the proper (or intrinsic or rest), mass of the particle, constant and independant of the observer, m0:

40

Special Relativiy

d m 0 v d m0 v = dt d The relativistic (or inertial or apparent) mass m, depending on the reference frame, is defined by the relation m0 m =  m0 = 2 v 1– c2 If one does not want to use the relativistic mass, one has to use the relativistic acceleration as defined earlier or to always replace the mass by m0. We will use here this French et Feynman notation, except, eventually, at low speeds where they are equal. According to this formula, the relativistic mass dilates with the same law as the time and increases indefinitely when the velocity tends to the speed of light. The Young’s double slits experiment with single photons shows the double nature, undulatory and corpuscular, of light. It is often asserted that the photon has no mass. This is of course true but only for the rest mass since, for v = c, the denominator of the above formula being zero, the numerator has also to be zero. The relativistic mass of the photon may be determined only from the equivalence of mass and energy and the quanta hypothesis. The variable mass is useful to allow a generalization of Newton' laws in the domain of relativistic velocities, near the speed of light.

4.3. RELATIVISTIC NEWTON'S SECOND LAW A REVOIR The Lorentz transformation of the accelerations, d v = dv dt d becomes, when multiplied by the rest mass m0 : d m 0 v d m0 v = dt d where is the proper time, in the mobile frame where the particle is at rest. v is the relative velocity of the observer to the particle reference frames. In the proper reference frame, the second Newton's law applies classically since the velocity is zero, thus low relative to the speed of light. One may then write : d m0 v d m 0 v d mv F= = = d dt dt The force having the same value in the observer's and in the particle reference frames, is therefore conserved in a change of frame, in one dimension of space

Special Relativiy

41

at least. In relativistic dynamics, the force, according to the relativistic Newton's second law is: F = dp dt where m0 v p= 2 1 v c2 is the momentum, product of relativistic mass and velocity. The proper time does not appear here any more since everything happens in the observer's frame. Let us take the example of a voyager moving away in a rocket and an observer remaining on the Earth. Both will be able to measure their relative acceleration with the help of an optical instrument like a laser velocimeter. The voyager will measure his acceleration with a mechanical accelerometer made of a load attached to a spring. With an identical instrument, the observer will measure the acceleration of gravity. Only the optical method will give the same relative acceleration for the observer and the voyager. In order to get the same result with the optical and mechanical measures, the voyager will have to subtract the acceleration of gravity, varying with the distance from the Earth. He will know the force from the ballistic caracteristics of the rocket. Now, what happens at relativistic speeds? As the rocket reaches the speed of light, the relative acceleration tends to zero but the proper acceleration may remain constant if the proper force is constant. In newtonian mechanics, will both measures give the same result at relativistic velocities? The optical method will give the constant velocity of light and therefore a null acceleration. The mechanical method will give the the assumed constant proper acceleration. The applied force may be known from the ballistic caracteristics of the rocket. The observer on Earth has no means to know the thrust. Anyway how to measure independently acceleration and force? The voyager measures his proper acceleration with the accelerometer and the relative acceleration with the laser. At the speed of light, he will be unable to measure anything. At a slightly lower speed, he will measure a Of course, Newton's law is valid in every frame but, in the proper frame, the acceleration relative to the observer is measurable with a mechanical accelerometer and an optical instrument. The observer on Earth is able to measure the acceleration with an optical instrument only. He is unable to measure the force. Another example is that of a lienarly accelerated electron. Only the accelerating potential (or the electrostatic field) and the velocity may be known.

Special Relativiy

42

est la quantité de mouvement fonction de la masse au repos m0 et de la vitesse v. Le temps propre n’apparaît plus ici car tout se passe dans le référentiel de l’observateur. One uses the letter a = dv/dt rather than  to designate the acceleration in order to avoid confusion with the Lorentz factor. Knowing that p = m0v, when the force derives from a potential V, one may write V d v F = m0 =– dt x v2 1– 2 c This is the same as using the relativistic mass m =  m0 with the classical acceleration

dv dt or the rest mass m0, invariable, with the relativistic acceleration d v dt Newton's second law of motion is relativity compatible if one takes into account the mass variation with velocity. It needs only to derive momentum instead of the velocity alone. Another method would be to consider the relativistic acceleration, not used.

4.4. ENERGIE CINÉTIQUE In classical mechanics, the kinetic energy is T = mv2. The velocity v, according to relativity, is limited by the speed of light. The maximum kinetic energy would be mc2 if the mass were independent of the velocity. It is a first approach of the relativistic energy. A second approach is to calculate the classical kinetic energy T with the relativistic mass: m0 m= v2 1– 2 c or, for m  m0: 2 2 m m T = mv = m c 1 – 0 1 + 0  m  m 0 c 2 2 2 m m The kinetic energy is proportional to the mass variation. We shall show that it is the relativistic formula. Let us apply the relativistic newtonian law. The variation dT of the kinetic energy being equal to the work of the applied force F

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43

during the displacement dx, we have, by applying the relativistic second newton's law: d v dT = F dx = F v dt = m 0 v dt = m 0 v d v dt Having the identity: v2 v d v = v d

v 2

v 1– 2 c

=

v dv 2

v 1– 2 c

+

v dv c2 2

1– v c2

3 2

=

v dv 2

v 1– 2 c

v2 c2 1+ 2 1 – v2 c

=

dv 2 2

21– v c2

3 2

=d

1 v2 1– 2 c

The incremental kinetic energy dT = m0 d may be integrated: T = m0 c2 + constant The constant is obtained by noticing that the kinetic energy must be zero at rest when  = 1. The constant is therefore - m0c2 and the kinetic energy:

T = m – m 0 c2 The relativistic kinetic energy is proportional to the mass difference between rest and motion.

4.5. E = MC2 The conversion of mass in energy had already being considered by Newton. Formulas like Einstein's had been proposed by Thomson, Heaviside et Poincaré. Lise Meitner used Einstein's theory to show that the mass lost during the fission of uranium was changed changed to energy. We shall derive, using the expression of the kinetic energy, T = (m – m0) c2, the most famous formula of modern physics. c and m0 being constant, the increase of the kinetic energy is due only to the increase of the relativistic mass m. In classical mechanics, the energy E is undetermined to an arbitrary additive constant E0. We may choose it such that E = T + E0 = m c2. The total energy in motion is then E = m0c2. At rest, v = 0, then  = 1 and E0 = m0c2. The rest energy is a constant for a particle at rest. Rather than choosing arbitrarily E0, one may call a evident principle. Indeed, the proportionality between mass and energy is well known in practice, for example by the car drivers. The energy contained in a given mass of fuel is proportional to it according to a coefficient K depending on its heat content. There should exist a maximum value of K corresponding to the maximum energy available when all the matter is transformed into pure energy. K should be a universal constant independent of the reference frame and from the velocity if mass and energy are equivalent. For a given object, the total energy will be: E=Km in the frame of the observer and

= d

44

Special Relativiy

E0 = K m0 in the prper frame of the object. The difference in these two energies is due only to the velocity: it is the kinetic energy: T = E  E 0 = K m – m0 K being a universal constant by assumption, only the mass depends on the speed. Now, the application of the second law of Newton combined with the definition of energy had shown that the kinetic energy was: T = m  m0 c2 Identifying these two las expressions, one finds K = c2 and, therefore, the total energy in motion or at rest is:

E = m c2 The Lorentz factor 2  = m = m c2 = E m0 E0 m0 c

represents the ratio of the total energy in motion to the total energy at rest as well as the ratio of the corresponding masses. The available energy in a particle depends on the observer e.g. if the particle is in motion or not relatively to the observer. This is not only true for relativistic velocities but also in classical mechanics. A car driver is often only aware of the damage he can cause at the time of a shock. The kinetic energy, even newtonian, is relative since it exists only relatively to an obstacle, that is, depends on the reference frame. All the derivations leading to E = mc2, need additive hypothesis In a few words we shall resume the reasoning conducting to this formula. Its origin is in the velocity of any material object limited to that of light. If a constant force is applied to the object to accelerate it, the velocity being limited and the force constant, it is necessary that the mass increases to avoid overcoming the velocity of light. The simplest formula giving an infinite mass for v = c is the dilatation of mass given by relativity: m0 m= v2 1– 2 c From this formula one gets the newtonian kinetic energy as a function of mass, approximated at low velocity but also valid for relativistic velocities: T = m – m0 c2 By assuming proportionality between mas and energy, one finds that the proportionality constant is c2. All the demonstrations using the transformation of matter into light or collisions need one or two supplementary assumptions. The hypothesis of proportionality

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45

of energy and matter with a universal constant seems better. The kinetic energy is thus proportional to the mass variation. Using the relativistic formula for the kinetic energy one obtains the value of the coefficient K = c2.

4.6. POTENTIAL ENERGY The variation dV of the potential energy is the product of force F and displacement dx with opposite sign. In the international system (SI), the potential energy is expressed in joules (J or N.m.). The energy units The second Newton's law gives the relationship between potental and kinetic energy. 2 d mv dV =  F x dx =  v dt =  d mv =  dT dt 2 V is the potential energy, not to be confused with the potential like the gravitational potential equal to the potential energy divided by the mass. The gravitational potential energy is always negative except eventually near the Earth's surface. The electrostatic potential is the potential energy divided by the electric charge. The sign of the electrostatic potential depends on the sign of the electric charge. The kinetic energy T is always positive. An adimensional potential is represented by the letter . The field is the derivative relative to space of the potential. The force is the space derivative of the potential energy. The potential disappears in special relativity, when switching from Newton to Einstein, reappears in relativistic dynamics as the Lagrangian "à la Landau" and in general relativity in the metrics, disappears again in the Einstein equations in the same way as in the gravitational or electrostatic Laplace equation. In classical mechanics, the total mechanical energy is the sum of the kinetic and potential energies. The conservation of energy is a consequence of Newton's laws and of the definition of energy. Conservation of energy is expressed by the relation T + V = constant expressing the relation between kinetic and potential energies. In special relativity, the total energy is E = mc2, without any reference to any potential energy. In relativistic dynamics, the conservation of energy could be written as T + V = m  m 0 c 2 + V = constant Using the definitions of the classical total mechanical energy and of the Lagrangian L: m 0 c2  m 0 c 2 + V = V0 v2 1 c2 We will encounter below the Lagrangians "à la Landau":

Special Relativiy

46

L =  m0 c2 1 

v2  V(x) c2

and in Newtonian limit of general relativity:

L=

d = dt d d

2 1 – v2 + 2V 2 = 1 c m0 c

both differing from the first one. This problem seems to be the clue of the incompatibility between special and general relativity.

4.7. ELECTRON ACCELERATION Energy The total mechanical energy, sum of kinetic T and potential energy V, is an arbitrary constant in the absence of dematerialization. In relativistic dynamics the kinetic energy being T = (m - m0) c2, the conservation of energy writes: m 0 c2 m 0 c2 2 – m0 c + V =  m 0 c 2 + V0 2 2 v 1 v 1  20 2 c c where V and v, V0 and v0, are respectively the potential and velocity at two different places in the physical space. We may write v0 = 0: m 0 c2  m 0 c 2 = V0  V = V v2 1 2 c or

v =c 1 1+

1 V

2

m 0 c2

V must be positive in order to have a real value of the velocity v. Therefore, the formula is not applicable to gravitation nor to an attractive electrostatic Coulomb force. The velocity tends asymptotically to the speed of light c when the potential difference increases indefinitely as is observed in particle accelerators. To check experimentally the formula, the velocity of the particle is measured as a function of the applied potential. The first measures were made in 1915 by Guye and Lavanchy measured in 1915 the ratio e/m in function of the velocity. Bertozzi, in 1964, measured the speeds of electrons with kinetic energies in the

Special Relativiy

47

range 0.5–15 MeV. The kinetic energy, determined by calorimetry,verifies that an electric field exerts a force on a moving electron in its direction of motion that is independent of its speed. Four experimental points seem to be insufficient. More precise measurements should be made. The Stanford linear accelerator (SLAC) is three kilometers long to accelerate electrons to 20 GeV with 82.650 one inch long accelerating structures divided in three cells. The accelerating voltage is thus less than 100 kV per stage, clearly less than 0.5 MeV, the total rest energy of the electron. The circular trajectory of cyclotrons and synchrotrons is obtained thanks to the magnetic part of the Lorentz force, perpendicular to the trajectory. The Lorentz and cetrifugal forces are in equilibrium (SI units): v2 q v B = m0 R where B is the magnetic induction, R the radius of the ring, me the rest mass and e the electric charge of the electron. In practice B and R have to be adjusted in function of the speed desired: mev BR= v2 e 1 2 c The magnetic field being limited by the power of the electro-magnets, the accelerators have an increasing size, like that of the CERN with a radius of 4 km.

Time Electrically charged particles are accelerated by an electrostatic field. We use here the word acceleration in the sense of increase of velocity, while it is increase in energy for accelerator specialists. It may be understood since a particle reaches the speed of light for relatively low energies, of the order of one MeV for an electron and one Gev for a proton. Let us apply the relativistic second Newton's law to an electron with a constant eletrostatic acceleration: d v =  dV = F = eE = g dt dx me me v2 1 2 c where e is the electric charge, E the electric field, me the mass of the electron and g the constant proper acceleration. The calculation, already seen, gives

Special Relativiy

48

c

v= 1+

mec

2

eEt The velocity of the electron tends asymptotically to the speed of light c.

4.8. RELATIVISTIC LAGRANGIAN "À LA LANDAU" In Minkowski space the motion is rectilinear and with constant speed. We shall determine the lagrangian of a particle subjected to a force deriving from a potential V : V F=– x The second Newton law is : d mv = – V dt x where m may vary with speed or some other variables. The time is the observer time. In relativistic dynamics we have m0 v V d =– 2 dt x v 1– c2 where m0 is the proper masse, constant. The expression in parentheses may be integrated :

d  – m c2 1 – v2 0 dt v c2

=–

V x

If the potential is independant of the velocity v, one may subtract it, on the left. On the right side one may add the derivative of the radical, independant on the abscissa x :

d  - m c2 1 - v2 – V 0 dt v c2

–

2  – m 0 c 2 1 – v2 – V = 0 x c

Let us define the lagrangian "à la Landau" as

v2 L =  m 0 c 1  2  V(x) c The preceding equation becomes the Lagrange equation : L L – d =0 x dt v 2

Special Relativiy

49

If V = 0, one gets the relativistic lagrangian of a free particle :

v2 L= 1– c2 The lagrangian "à la Landau" differs from the relativistic T – V : 1 T  V = m  m c2  V = m c2  1 V 0

0

v2 c2 There seems to be a problem, even if both lagrangians give the newtonian lagrangian at low velocity : 1 L  m 0 v 2  V x + constant 2 The lagrangian "à la Landau" is used in particle accelerators taking into account the electrostatic and magnetic potentials : 1

2 L = – m 0 c 2 1 – v – q (x) + q v • A c2 The potential V is replaced by q where q is the electrostatic charge and  the

electrostatic potential. A is the vector potential. The lagrangian "à la Landau" works for acceleration energies larger than the total rest energy of the accelerated particle. From the fundamental law of the relativistic dynamics we have obtained a "relativistic" lagrangian where the distinction between proper time and absolute time does not appear. This lagrangian is incompatible with Minkowski space and seems unable to predict any light deviation by the sun, contrarily to newtonian mechanics as we shall see in the following chapter dedicated to general relativity.

4.9. ANTIMATTER The total energy E = mc2 may be writen : 2 m 20 c 4 m 20 c 2 v 2 m 20 c 2 v 2 m 20 c 4 v 2 2 4 E =m c = + = 1 - 2 + mc 2 v 2 2 2 2 2 c 1 - v2 1 - v2 1 - v2 1 - v2 c c c c By replacing mv by the linear momentum p one obtains a useful relation, called dispersion relation between energy E and relativistic momentum p E 2 = m 20 c 4 + p 2 c 2 This formula works for a zero proper mass particle like a photon. The mass being squared, by taking its square root, there are two solutions with positive and negative masses :

50

Special Relativiy

E = ± m 20 c 4 + p 2 c 2 According to quantum mechanics also, there should exist negative masses called antimatter but the existence of negative masses has never been proved. When one speaks of antiparticles, it is about particles of the same mass but of opposite electrical charges. A photon and an antiphoton cannot be distinguished. The antineutron has been discovered in 1956 through its annihilation, but has not been observed directly.

Special Relativiy

51

5. CONCLUSION ON SPECIAL RELATIVITY The unsolvable problems encoutered at the end of the 19th centuryhave been clarified by Einstein with his special relativity. He has rederived the Lorentz transformation with a different basis. He modified the classical mechanics by taking again the Galilean principle of relativity abused by Newton with his absolute time and space. The Galilean transformation is replaced by that of Lorentz, so that speed and acceleration are not any more simple derivatives of space with respect to time. The speed of electromagnetic waves is that of light and depends only of electric and magnetic properties of matter measured in the laboratory. The electromagnetic wave equation does not depend on any absolute reference frame, contrarily to mechanical waves. The light wave is insensitive to the wind even of Aether. The Michelson experiment did not give the result predicted by the Newtonian mechanics, even with the use of extra-terrestrial light. Lorentz and Fitzgerald invented time dilatation and length contraction. Stellar aberration, pi-ion experiment, double star Algol, none of them contradicts the constancy of light speed, at least in the absence of gravitation. Superluminal velocities of so-called tachyons would have been observed but have been explained by a perspective effect. The measure of mesons lifetimes, the Fizeau experiment and the Doppler effect are quantitatives verifications of the Lorentz factor and of the slowing down of the time. The relativistic dynamics, useful in practice, is a generalization of the newtonian dynamics. Adding the hypothesis of proportionality between energy and mass leads to the well known formula E = mc2. The relativistic lagrangian "à la Landau" is equivalent to the relativistic Newton's second law, useful in the particle accelerators, but ineffective for gravitation. Therefore, the theory is incomplete, as compared to rational mechanics valid in electrostatics and gravitation although not at speeds near that of light.