Let's generalize theabove to velocities alongdifferent directions. Using the notations of the previous section, we now considerthe following motion in the frame of reference attachedto an actual particle moving at speed  v < c. 

x' = u t' cos         and        y' = u t' sin

Using the expressions of  x'  and  t'  given by theLorentz transform (along with  y' = y)  we obtain the following description in termsof the observer's coordinate system:

x vt   =  u ( t xv/c²) cos         and        y   =   u ( t xv/c²) sin

From the first equation, a relation of the form x = w_x t  is derived just as in thecollinear case.  For the other coordinate,we may start by remarking that:

y   =   w_y t  =   (w_x - v) (tg ) t

This leads to a straightforward computation of  w_y which simplifies nicely:

Cartesian Components of the Observed Resultant Velocity

	wx   =	v + u cos		wy   =	u (1 v2/c2)½sin
		1  +  (uv cos ) / c2			1  +  (uv cos ) / c2

The relation  w² = (w_x)² + (w_y)²  gives the speed  w  seen by the observer.  With some algebraic massaging, thiscan be put in a nice symmetrical form:

	1 w2/c2  =	( 1 u2/c2)( 1 v2/c2)
		[ 1  +  (uv cos ) / c2] 2

The reader may want to retrieve the aboveaddition formula for parallel velocitiesby putting cos  = 1 in this lesser-known general relation.

In the case  u = c,  we have w = c  and the above expression for w_x  turns into the following relativistic formula, published by Einstein in 1905.

Relativistic Aberration of Light

	wx	=   cos   =	(v/c) + cos
	c		1  +  (v/c) cos

In its rest frame, an isotropic source of photons is equallylikely to emit at an acute angle or at an obtuse anglewith respect to any direction of reference.

That direction of reference can be chosen to be thevelocity v  of the source with respect to a particular observer,who will thus see half of the photons emitted within a conecorresponding to a right angle in the rest frame of the source. Using the results of the previous section (with  u = c  and sin  = 1) the limiting cone is defined by the following components,respectively parallel and perpendicular to v in the frame of the observer:

w_x   =   v        and        w_y   =  c /

This corresponds to an angle   =  Arcsin (1/) =  Arccos (v/c) away from the direction of the velocity v. For a fast source,   is small, which indicates that the photons aremostly emitted in a direction close tothat of v.

Two spaceships head toward each other at 70% and 80% of the speed of light,respectively. They are 3 light-years apart.  When will they meet?

Well, the distance between them is seen to decrease at a rate equal to 0.7 + 0.8 = 1.5  times the speed of light. They'll meet in exactly  2 years.

Relative Speed  vs.  Closing Speed

Thepreviously discussed relative speed  of two moving objectsis defined as thespeed of one object seen by an observer at rest with respect to the other.

This is entirely different from what's sometimes called the closing speed,  which is the rate of change of the distancebetween two moving objects  (as seen by an observer who is linked to neither).

When motion along a straight line is considered at time t by an independent observer,an object moving at velocity  -u  is at abscissa  -u t  andan object moving at velocity  v  is at abscissa  v t. The signed distance separating the latter from the former is clearly  (u+v)t and the rate of change of that quantity (the signed closing speed,  if you will) is  u+v. In prerelativistic mechanics, there is no difference between the relative speed and theclosing speed of two objects  (because two moving observers are supposedlyexperiencing the same flow of time). In relativistic mechanics, this ain't so.

Confusing the two notions can be a great source of puzzlementwhen the relevant conditions are not properly analyzed. In particular, the followingFizeau effect is correctlyexplained by the relativistic expression for relative velocities, whereastheSagnac effect is due to the difference in the closingspeeds of two light beams that either chase a moving mirror or race toward it (those closing speeds are  c-v  and  c+v,  respectively).

In a transparent fluid at rest,the  [phase] celerity of light  u = c / n  is isotropic andinversely proportional to the  fluid'sindex of refraction  (n).

Consider the case where the propagation of light is parallelto the motion of the fluid. Let v be the speed of the fluidand w the observed  celerity of light. According tothe above rule, we have:

The parameter  f = 11/n²  is known as the Fresnel drag coefficient. It has been named after Augustin Fresnel (1788-1827) who introduced the now obsoleteaetherdrag hypothesis in 1818, to explain experiments performed byArago in 1810. The coefficient f  would be 0  if the motion of the liquid had no influence on the propagation of light. It would be  1  if light was entirely "carried" by the liquid,like sound is. What's observed looks like partialdragging.

Hippolyte Fizeau established empirically the above expression of  f  in terms of the index n, by experimenting with different liquids. Although Fizeau's relation can be derived without resorting to the principleof relativity  (Lorentz did it) Einstein considered it an excellent experimental test of Special Relativity.

The Sagnac effect is mainly a nonrelativistic  effect observedwhen two coherent light beams  (which may come from a single source througha ) travel in opposite directions around an optical loop. When the apparatus rotates, a phase difference is observed which is essentiallyproportional to both the pulsatance   [the rotation rate in rad/s] and the area of the loop  (actually,the  apparent area, for a distant observer locatedon the axis of rotation).

The loop may be a polygonal path, with a mirror at every corner (light travels at speed  c  between mirrors). Alternately, fiber optics may be used so light travels at a celerity  c/n relative to the loop  [ n being the refractive index of the optical material ]  via ordinary total internal reflection  (TIR).

 10 ^-11 rad / s  have actually been detected this way...

We consider only the case of a circular loop, of radius  R,rotating about its axis (note that  n = 1  approximates a regular polygonal path withmany mirrors).

Let's measure positively counterclockwise angles  (that's the usual convention) and assume that the loop is rotating in this positive direction. Euclidean geometry remains valid for a fixed observer, who thus sees each pointof the loop travel a distance  2R in a time  2. Therefore, the nonrelativistic expression for the speed  v of each point of the loop does hold:  v = R. (TheFitzGerald-Lorentz contractionapplies to the length of moving objects,notto the space they travel.)

Using this value of  v  in the (exact) expression from theprevious article, we obtain thevalue for the celerity  w  of either beam in the moving loop,in our fixed viewpoint ("" means"+" for the positive beam and "" for the other).

w    =     c/n  +   R( 11/n²) [ 1 R/nc ] ¹

This signed quantity is, of course, equal to  c when  n = 1. Now, if both beams start from the same point at  t = 0, this point will be at an angle  t when the beam reaches it again after one turn, so that  t  is a solutionof:

t w / R    =    t

Solving for  t,  we get:  t  =  2R /( -wR )   t⁺  =  2R /( w⁺ R )
The exact  values of w make  n  vanish from the time lag !   [2005-07-27]

Sagnac Time Lag  (inertial observer at rest at the center of rotation)

t+ t    =    t    =      4 R 2 c 2 2 R 2

With ordinary objects, the above is indistinguishable from 4 R ² / c² and is thus proportional to the rotation . Indeed, the speed  R  of the circumference is much  lower than thespeed of light (the relative error is about 1.22  for a 10 cm radius at 10 000 rpm). Even for relativistic things, the above denominator must be positive (there's no such thing as a large  rotating "solid").

A Brief History of the Harress-Sagnac Effect :

The Sagnac effect was first dreamt of byOliver Lodge (1851-1940) in 1893 and byAlbert A. Michelson in 1904.

In 1911, Francis Harress tried to substitute glass for theliquids used inFizeau's investigations. He had the idea to observe rotating rings of glass, but could clearly do soonly at fixed points of such rings... As we've discovered theoretically in the above discussion, the resulting effectdoes not  depend on the index of refraction involved ! Although Harress failed to understand his experimental results,the effect is still known as Harress-Sagnac, mostly in the context of fiber optics.

In 1913, the French physicist Georges Sagnac (1869-1926) published his ownexperimental results and properly described the effect now named after him. The prior observation by Harress of the effect's "fiber optics version"was noted later.

Michelson and Gale used this effect in 1925 to measure the absolute rotationof the Earth, with a rectangular optical loop 0.2 mile wide and 0.4 mile long.

Answer:  The quantity w doesn't have a definite limit as both u and v approach c.

This formula for w describes how [collinear] relativistic velocities are combined: Consider a straight railroad track where a train (U) moves at speed uand another train (V) moves at speed v(both speeds being measured relative to some platformon which the "observer" is located). According to the Special Theory of Relativity (seeabove)  the quantity w = (u-v)/(1uv/c) is simply what the speed of train U would be for an observer located on train V.

If the observation platform is on a fast rocket movingparallel to the railroad tracks and approaching the speed of light c(with respect to the tracks),both u and v will be close to c. Yet, the quantity w remains a low number (like 10 mph or 20 mph)equal to the speed of one train relative to the other. This relative speed is not affected by how fast an irrelevant nearby rocketmight be moving...

To make this relativistic math more transparent, you may want to considerrapidity instead of speed. The interesting thing is that rapidities areadditive,whereas speeds are not: The rapidities x, y, z corresponding to the above speeds u, v, w are thus related bythe much simpler equation:

z   =   x - y

As speeds approach c,rapidities approach infinity and the question becomes:

What's the limit of  z = x-y when both x and y become infinite?

Answer:  Such a limit is clearly undefined.

Take your pick, use either the physical or the mathematical approach...

The velocity of object B relative to object A is the velocity of B measured in the frame ofreference where A has zero speed. When A is a photon,we are in trouble with this definition,because there is no such proper frame of reference.

The question is thus to determine if/when the above definitioncan be consistently extended to include objects moving at speedc. Such an extension exists, except in the very special case of two photons movinginexactly the same direction, where the notion of "relative speed"breaks down completely(as shown in theabove discussion of the relativisticformula for the "addition" of parallel velocities).

If A is a photon but B isnot also moving at speedc,we may still reach a firm conclusion by noticingthat the velocity of B relative to A must be the opposite of the velocity ofA relative to B (which is well-defined, unless B moves at speedc).

Let's consider now the case of two photons of velocitiesu andv. These two velocities are 3D vectors of lengthc; the corresponding 4-vectors(c,u) and (c,v) have zero 4-dimensional "length". If u andv are not equal (that's to say that the two photonshave different directions) the 3D length||uv|| is nonzeroand it turns out that the relative velocity of two objects chasing thetwo photons (at sub-c speed) approaches a definite limit when the velocitiesof the chasers both approach the velocities of their respectivechasees. This limit is:

c (uv) / ||uv||

If we have to define the relative velocity of two photons moving along differentdirections (possibly opposite, but not equal),this is the only sensible way to do it.

On the other hand, ifu andv are equal, we are back to ourprevious discussion: Two sub-c chasers with anyarbitrary relative velocity(not necessarily collinear withu = v)could both approach the velocity (u = v) of both photons. Therefore, there is nocontinuous way to define the relative velocities of two photonsmoving in the same direction. Having said this, we may or may not find it useful to state(ratherarbitrarily) that two such photons have zero relative speed. However, we failed to find a compelling reason for this "obvious" choiceand cannot guarantee that it would not lead to a paradox of some kind...

(2005-04-14)  The Lorentz transform  actually expresses the rule fororthogonal coordinate transformations in  4D spacetime. Space by itself and time by itself are fading into mere shadows.  Only a union of the two will preserve an independent reality. Hermann Minkowski  (1908)

Minkowski went on to explain that a timelike component is associated withany legitimate physical 3-vector which makes the [contravariant] coordinatesof the resulting 4-dimensional object transformaccording to the very same Lorenz transform, introduced above for positional coordinates.

A scalar quantity    whose value does not depend on thecoordinate system used to locatespace-time events is called a relativistic invariant. The 4-dimensional gradient of such a scalar is the following 4-vector:

Grad   =  (-1/c t ,grad )   =  (-1/c t ,x ,y ,z )

Applying this definition in another coordinate system leads to coordinatesthat are indeed obtained from the above ones via the relevant Lorentz transform.

(2006-03-28) 
A Lorentz boost  of speed  V = c  may be expressedvectorially.

Let V  be the [vectorial] 3-dimensional speed of(S') relative to the coordinate system (S). Introducing the vectorial  quantity  = V/c,we may remark that any 3D vector A  is the sum oftwo vectors, one parallel to  (theprojection onto ,expressed with a dot product ) the other perpendicular to it.

(A./)/        and        A   (A./)/

TheLorentz transform applied to the 4-vector (a,A)  doesn't change the latter and specifies the interrelatedtransformations of the time coordinate  (a) and of the "parallel" spatial coordinate (A./). Putting it all back together, we obtain:

a'  =    (a   A. ) A' = A  + [ (1)A./2   ]

A linear  transformation which preservespacetime intervals  (while respecting the orientation of space andthe direction of time)  is necessarily a composition of such a boost  with a spatial rotation (which leaves time unchanged and preserves spatial distances withoutchanging the orientation of space). Such transformations form the  6-dimensional Restricted Lorentz Group (a spatial rotation may be specified by its 3 Euler angles, whilea boost may be given by the 3 components of the vector   introduced above).

If space inversion and time reversal are allowed, we obtain theLorentz GroupSL(2,C) with  4  connected components isomorphic to the restricted  group.

(2005-04-16) 
An Introduction to Four-Dimensional Wave Vectors.

The phase   of a periodic phenomenon describesits position in the cycle:  a crest, descending, a trough, ascending, etc. This does not depend on whatever coordinate system we may use to locate the event:   is a relativisticinvariant.

The phase    of an ideal planar wave  at a given point R = (ct,r)  is given by the following relation,up to some irrelevant additive constant:

  =   tk.r

Thepulsatance  ()  is proportional to the wave'sfrequency  (), whereas the magnitude of the 3Dwave-vector  (k)  is tied tothe wavelength  (), namely:

	=	2
||k ||	=	2

Thecelerity of the wave  (itsphase speed)  is the product u = .

K = (/c,k)  is a4-vector because it's the4D gradient of a scalar invariant:

K   =  (/c ,k)   =  Grad(-)

This quadrivector is known as the four-dimensional wave vector. The 3-D vector k  is called either wave vector or  (less often) propagation vector.

(2005-05-03)  
The radial effect is multiplied by anisotropic relativistic factor.

Amongredshift factors, the classical Doppler effect  comes from a changing distance between theobserver and the source of radiation. Another  type of redshift is also due to local relative motion. It's entirely relativistic and applies even to transverse  motion, when this distance doesn't change:

If the source is moving at velocity v, its proper pulsatance ₀ = 2₀ (in its own rest frame)  is given by theLorentz transform for thewave-vector:

₀/c =  ( /c    k _x ) =  ( /c   k.v/c )

Therefore, calling    the angle betweenv andthe direction of observation (-k) we have:

₀ =  (  +  cos() ||v|| / )

If the signal propagates at celerity u =  (not necessarily c)  in the frame of the observer, we obtain the followingrelation  (seenote below if  u<c):

n0   1 v/c   =    u + v cos

This is merely the classical (radial) Doppler effect,with an extra relativistic factorcorresponding to the observed stretching of time  in a moving source.

The above is of the form  ₀' = u' :  The classical radial celerity (u') isthe wavelength ()multiplied into a relativistically adjusted frequency(₀').

A Dubious Academic Tradition:

Things become more obscure  when the above is "simplified" for light ina vacuum  (u = c) in 3 special cases which are popular with textbook writers:

Note, when  u < c :

As remarked in the general derivation, the above is only valid when thesignal emitted by the source obeys a classical wave equation with celerity  u in a frame at rest withrespect to the observer.

For light in a vacuum, this is of no concern because the maintenet of special relativity does state that somethingthat propagates as a wave of celerity  c  in one particularinertial frame does so in any other inertial frame as well.

Not so when the celerity  u  is less than  c,  though! In that case, the wave equation is only valid in the proper frame ofthe propagation medium  (e.g.,  air in the case of sound). Otherwise, we are faced witha complicated combination of the Doppler effect  and the Fizeau effect (only the latter is at work when the source and the observer move at the samevelocity with respect to the propagation medium, as when one listens toan outdoor concert in a steady wind).

(2005-04-14) 
E/c andP form a 4-vector  (i.e., they transform like ct andr).

So far, we have only dealt with relativistic kinematics by introducing quantities on which a description of motion is basedwhich is consistent with the basic tenets of Special Relativity,namely equivalence of all observers in relative uniform motion andconstancy of the speed of light measured by all such observers.

Another principle  has to be introduced to provide thephilosophical equivalent of the basic laws of Newtonian mechanics which introducethe notion of force  and relate it to changes in motion...

One approach of Newtonian mechanics which can be consistently generalizedto the framework of Special Relativity  is to introducethe notion of linear momentum  (the product P of mass  m  by velocity v) and to postulate that the time derivative of that quantityis a vectorial quantity, called force, which describes dynamicalexchanges between distinct parts.  (This is where Newton's famous equation  F  =  m a   comes from.)

Newton postulated that "to every action,there's an opposite reaction of the same magnitude"  which is afancy way of saying that every variation in the momentumof one part will always be exactly compensated by the variation in themomentum of another, so that total momentum is conserved.

E2   =   (m c2) 2  +  (p c) 2

Louis Vlemincq(Belgium; e-mail 2005-04-14) 
What's the relation between  E = m c ² and the formula  E = ½ m v ² ?

The relativistic energy of a particle of rest mass m  and speed  v  is:

E   =  (m )  c 2   =	m c 2
	1 - v2/c2

  as we reservethe symbol  m  itself for the so-called "invariant mass"or "rest mass" of a particle  (followingcurrent standardpractice).

Under the assumption that v is much smaller that c, a good approximation for Eis obtained from the Taylor expansion of the above:

E   =   mc ²  ( 1 + ¹/₂ (v/c)² + ³/₈ (v/c)⁴ + ⁵/₁₆ (v/c)⁶ +  ...   )

At low speed, we may just keep the first two terms of this series:

E    mc ²  + ½ mv ²

In prerelativistic mechanics, the first term is irrelevant because it's a constant, whereasthe second term is called kinetic energy. Actually, the first term was originally conjectured by Einstein because of the aboveconsiderations:  It's just mathematically simpler to assume thata motionless body of mass m already has an energy mc. This explains painlessly the so-called mass defect  observedin the decay of radioactiveelements : A nuclear decay always leaves remnants whoserest masses add up to less than the massof the original nucleus.  The balance of the energy appears in the formof either kinetic energy or radiation...

The relation   E  =  m c ²   has been verified directlyby countless experiments in nuclear physics, starting (in 1932) with the artificialtransmutation of lithium into two alpha particles, using a beam of fast protonsproduced by the particle accelerator of Cockcroft and Walton (Nobel 1951).

In 1938, Otto Hahn and Fritz Strassmann observed that Bariumis produced when Uranium is bombarded with neutrons. In 1939,LiseMeitner and Otto Frisch interpreted this result as an induced fission  of the Uranium atom.

(2005-04-14) 
To have a finite energy, a massless particle must travel at speed  c.

Particles traveling at speed c can only have a finite energy if they havezerorest mass ;  they only exist in motion (always at speed c). In this case, we must use a quantum expression of the energy, in terms of an associatedwave:  The energy of a photon of frequency   is  h,where h isPlanck's constant. This relation was proposed by Einstein (1905) in his explanation of the laws ofthe photoelectric effect  (for which he receivedtheNobel Prize in 1921) which may be construed as a formal discovery  of the photon...

A generalization to electrons and other subluminal  particleswas proposed by the French physicistLouis de Broglie,who put forth anew principle establishingthedual corpuscular and undulatory natureof everything, as discussednext.

(2005-04-16) 
Corpuscular and undulatory duality, as proposed in 1923/1924. 

In 1923, the French physicistLouis de Broglie (1892-1987;Nobel 1929) was still a graduatestudent at the Sorbonne when he proposed the idea ofmatter waves,which he defended successfully in 1924  (with the support of Einstein himself) in front of a doctoral committee which includedPaul Langevin (1872-1946). At the time,de Broglie stated that his proposedmatter waves mightbe observable in experiments involving crystal diffraction with electrons. Suchexperimental confirmations came in 1927, with two independent experiments: one byClinton J. Davisson (1881-1958;Nobel 1937)andLester H. Germer (1896-1971), the other byG.P. Thomson (1892-1975;Nobel 1937)... Ironically,George Paget Thomson thus demonstrated theundulatory nature ofelectrons, whosecorpuscular properties were established three decades earlierby his own father,J.J. Thomson(1856-1940;Nobel 1906).

De Broglie's idea was that any particle is associated with a so-calledpilot wave: The momentum of one and the wave-vector of the other are proportional  andthe coefficient of proportionality is a universal constant. We'll statede Broglie's principleusing the4-dimensional wave-vector introducedabove:

K	=	(/c ,k)   =  Grad(-)
	=	2
||k ||	=	2

De Broglie's Principle Expressed Relativistically :

Louis de Broglie proposed that any particle of 4D momentum P = (E/c, p)was "associated with" a wave of(4D) wave-vectorK proportional toP, namely:

P   =  K   =  (h/2)K      [ h isPlanck's constant ]

This 4D equality breaks down into a scalar component and a (3D) vectorial component. The former is Planck's relation, the latter is de Broglie's relation,usually stated using only the magnitude p =  || p ||   of the 3D-momentum:

E	=	h
p	=	h /

Phase Celerity  (u) vs.  Mechanical Speed  (v)

The above relations make thecelerity  u =  equal to E/p. For a particle ofrest mass m and speed v, we have:  (mc)²  = (E/c)² p ².  Therefore:

1 / u ²  =  (p/E)²  =  1 / c ² (mc/E)²  =  (1-(1-(v/c)²)) / c ²  =  (v/c ²) ²

This establishes an extremely simple relation betweencelerity andspeed:

u v   =  c 2

The [phase]celerity (u) for a massive particle is thusgreater than c,as energy and information travel at thespeed (v),which remains lower than c.

De Broglie wavelength of a relativistic particle :

- (mc) ²   =   -(E/c)²  +  p ²  =   -(E/c)²  + (h/) ²      Therefore:

=    h c (E mc2) (E + mc2)

The relativistic kinetic energy  W  =  E  m c² is often used to characterize the speed of fast particles, especiallywhen  x = mc²/W  is small.

=    h c W   1 + 2x

On the other hand, in the nonrelativistic case where W  isapproximatelyequal to  ½ m v ² , we have   h/mv. More precisely, the usual expression of the 3D relativistic momentum (p)  yields:

=   h / p  =   (h / mv)  1 (v/c)2

Finally, here's a formula involving the Compton wavelength (h/mc)  :

=   (h/mc)  [ (E/mc2) 2 1 ] ½

(2023-03-03) 
All components obey the same second-order differential equation.

Involutive Fourier Transform

(2005-04-14) 
The frequency of a photon changes when it collides with an electron (ofrest mass m).

Compton scattering  is a deflection of  X-rays by matter that entailsa shift in their frequency() which depends on the angle ofdeflection ().

This was explained by Arthur H. Compton (1892-1962; Nobel 1927) in terms of collisions between incoming photons and recoiling electrons:

In a system where the target electron (of rest mass m) is at rest,its 4-momentum is  (mc,0). The incoming photon moves along the x-axis and goes outin the (x,y) plane, at an angle   from Ox.  Now,energy-momentum is conserved: After the shock, the 4-momentum P of the electron is thus obtained by subtracting the momentum of the outgoing photonfrom the sum of the two initial  4-momenta.

TheMinkowski square of the electron's 4-momentumis always  -(mc)².  So:

m ² c ⁴   =  (mc² + hh' ) ²  + ( h - h'cos ) ²  + ( h' sin ) ²

Using the Compton frequency of the electron (_c = m c² / h ) this gives:

'  =   / [ 1 + (1 cos ) / c ]

The effect is often stated in terms of a change in wavelength,using the Compton wavelength of the electron _c :

_c = h / mc = 0.0024263102389 nm

'  =    [ 1 + (1 cos )c /  ]

'   =  _c(1 cos )

The Compton shift (z) is best defined, like other typesof redshift,  as the relative change in wavelength. If  E  is the energy of the incoming photon, the outgoing photon has energy E/(1+z).

z   =  (1 cos ) / c  =  (1 cos )E / mc2

The recoiling electron, initially at rest, is imparted a speed  v  and a kineticenergy  W  equal to the opposite of the change in energy of the photon

W   =  E   =  E E/(1+z)   =   E /(1+1/z)

With energetic photons  (gamma rays) the target electrons may recoil at high speed andcause bluishcherenkov radiation in transparent bodies.

(2008-01-31) 
Differential cross-section of electrons in Compton diffusion.

Photons deflected by an angle between and +dspan asolid angle d = 2 sin  d. Such a deflection occurs with the same probability as would a collision of aclassical point particlewith an obstacle of cross section  d. ForCompton diffusion,this is given by the Klein-Nishina formula,as derived from a 1928 paper by Oskar Klein (1894-1977) and Yoshio Nishina(1890-1951) :

d    =   ½ (r_e) ²  [P    P² sin²  +  P³ ]  d
where   P   =  '/   =  [ 1 + (1 cos ) / _c] ^-1

This involves the classicalelectron radius,  obtained by equating the rest energy mc ²   with twice  the electrical energyof a sphere of radius  r_e  bearing the charge of the electron (q)  uniformly distributed on its surface:

re   =	q2	=  2.8179402894(58) fm
	4o mc2

The total cross-section   is obtained by integrating the above,  using the parameter u = /_c and a new variable  x = cos  :

=   (re) 2		Log (1+2u)	2	(u+1) Log (1+2u) 2u	+  2	u+1
		u		u3		(1+2u)2

For photons of low energies  (small values of  u) this total cross-section reduces to that of classicalThomson scattering, namely:

_o   =   ⁸/₃  (r_e) ²

At high energies, the cross-section () vanishes  butthe average transfer of energyincreases logarithmically with the energy of the incoming photon.

Bert Dobbelaere (2008-01-29; e-mail) 
Compton diffusion does not cause any optical aberration. Why?

Compton diffusion transfers some energy to the recoiling electron. With incoming gamma rays, this transfer of energy can easily exceed what'snecessary to overcome the binding energy of electrons bound to atomic orbitals. Beyond that threshold, a continuous spectrum of energy is allowed.

On the other hand, the low-energy photons of visible light can only transfersomething like 1/100000 of their energy to the electron they collide with,according to theabove formula (the maximum occurs when the photon bounces straight backto the direction where it came from).

The electrons bound in the ordinary orbitals of chemical elementscan only change their energies in steps which are far greater than that. Therefore, the recoil must be absorbed by the whole atomic structure rather than bya lone electron.

This reduces the relative Compton shift of low-energyphotons so drastically that no measurable effect can be observed. Even after many billions of wavelengths traveled through anoptical instrument, Compton diffusion at differentangles will cause no observable difference in phase. The Compton effect is thus completely suppressed quantically for visible light.

(2003-11-15) 
A simple relation between transfer of energy and change in momentum.

E   =  v.p (v is the velocity of the center of mass).  Here are the details:

If two particles (1 and 2) collide but retain their respective identities,we may define the energy and the momentum lost by one and gained by the other:

E		=	E'1 E1	=	( E'2 E2)	=	y c
p		=	p'1p1	=	(p'2p2)	=	xu     [ with  ||u|| = 1 ]

The scalars x and y are introduced for convenience, so is the unit vectoru,which is uniquely defined, unless no shock takes place(p = 0). We also introduce:

p_i  =  u . p_i  =  (E_i / c ²) v_i

A collision is said to be elastic when each rest mass is conserved, namely:

E'12 / c2p'12	=	E12 / c2p12
E'22 / c2p'22	=	E22 / c2p22

In the form  (p'_ip_i).(p'_i+p_i) = (E'_i E_i)(E'_i+ E_i) / c ²  these two read:

x ( 2 p1 + x )	=	y ( 2 E1 / c  +  y )
x ( 2 p2 x )	=	y ( 2 E2 / c    y )

Adding both equations, we obtain  x ( p₁ + p₂)  = y ( E₁ + E₂) / c Using this relation, we may multiply either of the previous equations by  ( E₁ + E₂) / cx (as x is nonzero) and obtain the second equation of the followinglinear system:

x ( p1 + p2)		y ( E1  +  E2) / c	=	0
x ( E1  +  E2) / c		y ( p1 + p2)	=	2 ( E1p2 E2p1)

By itself, the first equation already  says that  E   =  v.p (as advertised)  where the vector v  is the velocity of the center of mass, namely:

v   =  (p₁ + p₂)  c ²/ ( E₁+ E₂)

By solving the whole system for  x  and  y,  we obtain:

(2008-08-28) 
A process whose leading Feynman diagram is at right:

Classically, light does not interfere with itself becauseMaxwell's equations  are linear : Light beams normally pass right through each other undisturbed.

However, there are high-energy quantum processeswhose net result is similar to an elastic collision between twophotons  (cf. above Feynman diagram)  and we candetermine the relations between the incoming and outgoing photons.

If the photons are traveling in exactly  the samedirection, they will never collide  (just as cars which travel atthe same speed in the same direction of the same road do not collide).

Otherwise, their combined energy-momentum is a subluminal  4-vector,which is the energy-momentumof what can be called the center of mass of those two photons.  The mass of that point is the Minkowski-lengthof the momentum-energy divided by c. Its  3D-velocity is the 3D-momentum divided by that mass.

Let's use this center of mass asthe origin of a new frame of reference. In that frame of reference, the two incoming photons simplyhave the same frequency and opposite directions (so that their combined 3D-momentum is zero).  So do thetwo outgoing photons.  Furthermore, the incoming and outgoing frequenciesare identical  (because energy is conserved).

Usually however, the incoming and outgoing directions are different. In particular, there is an angle   between them, which departs from a zeroor flat angle whenever the collision breaks the axial symmetryof the incoming photons.

The azimuthal orientation of the outgoing direction in which the incoming symmetryis broken will also be important to whoever might have to use the relevantLorentz transform to translate the abovesimplicity into actual laboratory measurements.

For the head-on collision of two photonsof frequencies₀ and₁, we may write the conservationof energy-momentum  (divided by h/c)  as follows:

0  0   0  0

+

1  -1   0  0

=

0'  0'  cos0   0'  sin0  0

+

1'  -1'  cos1   -1'  sin1  0

(2005-05-05) 
The Cherenkov effect occurs when a charged particlemoves faster than the celerity of light.

In a dispersive medium like a liquid or a gas, the celerity of light dependson frequency.  Normally, the celerity of visible light is belowEinstein's constant (c) while the celerity ofX-rays is above it. ( It's thegroup speed which may never exceed c. Phase celerity is another matter entirely.)

Radioactive sources make nearby transparent bodies emit a bluish glow whichMarie Curie firstnoticed in 1910, with radium salts in distilled water. Only part of this glow consists of ordinary luminescence from impurities...

However, this is what the whole thing was mistaken for until the French radiologistLucien I. Mallet (1885-1981)studied the phenomenon in details in 1926-1929 and found it to have a continuous  spectrum, unlike fluorescence.

This was further investigated between 1934 and 1937 by Pavel A. Cherenkov (1904-1990) who established that the radiation came primarily from fast electronsdislodged byCompton collisions with energetic gamma-rays. The final mathematical explanation was worked out by two of his colleagues from the Lebedev Physical Institute  of Moscow, Il'ja M. Frank andIgor Y. Tamm, following the 1936 discovery of the particular geometryof the Cherenkov beam. For this, those three men became the first Russians ever to be awarded theNobel Prize in Physics,in 1958.

Cherenkov radiation is entirely different from so-called bremsstrahlung,the electromagnetic radiation emitted when a charged particle is accelerated (e.g., as it collides with atoms). A heavy particle causes less bremsstrahlung  thana lighter one of the same speed, but the Cherenkov emission is the same. The Cherenkov effect is to light what the sonic boom  is to sound.  Kind of :

The effect is also called "Cerenkov-Mallet", especially in French texts. "Cerenkov" and "Cherenkov" are equally acceptable transliterations.

Alexander  (Yahoo!2007-08-05) 
How far would you travel in a lifetime at a constant acceleration  g ?

Relativistically,a constant acceleration cannot be maintained indefinitely relative to a fixed female  observer  (Alice)  or else the speed of the male  traveler  (Bob)  would eventually exceed the speed oflight, which is absurd. Thus, the "constant acceleration" we're talking about is the acceleration he feels,  not the acceleration she sees.

Although the rest frame of Bob is not inertial at all,we may consider, at some specific instant, the so-called tangent frame  (S')which is an inertial frame that moves uniformly with respect to the frame of Alice (S)at the same velocity as Bob. When Bob uses the inertial frame (S') to describe his own motion, he finds the secondderivative of his position (x') with respect to time (t') to be constant (g).

(c. 2010)  
A puzzling effect confirmed by the Häfele-Keating experiment (1971).

In October 1971, Joseph C. Häfele (Department of Physics, Washington University, St. Louis, Missouri) and Richard E. Keating (Time Service Division, U.S. Naval Observatory, Washington, DC)conducted an experiment with four different cesium atomic clocks flownaround the Earth in two opposite direction (Eastward and Westward). The entire experiment lasted 636 hours, including a  65.4 h trip eastward and an  80.4 h  trip westward.

The double-twin paradox

As first stated by Langevin, the twin paradox arose when one twin wenttraveling while the other didn't.  The proposed solutionwas that the situation wasn't entirely symmetrical because the travelingtwin had to be subject to accelerations the stationary twin neverexperienced:  For example, the traveling twin starts by by firing rockets,cruises at constant speed for a while, fires rockets to turn around,cruises back at constant speed and uses the rocket one last time toarrive home at zero speed.  It's only then that both twinscan compare their clocks at rest in the same frame of reference.

Well, let's send both twins away in identical spaceships withidentical rockets.  Except that one of them turns around sooner.  What gives?

(2018-04-03)     (Lampa, 1924.  Terrell, 1957.)
Also called Lampa-Terrell-Penrose effect  or Penrose-Terrell effect.

This is a relativistic optical illusion which changes the aspect of fast-moving objects (in particular, moving spheres retain a perfectly round appearance in spite of theFitzHerald-Lorentz contraction).

That was first described in 1924 by the Austrian physicist Anton Lampa. However,  his published discovery was utterly ignored for decades.

The effect was rediscovered in 1957 by Jim Terrell (Nelson James Terrell Jr., 1923-2009) whose paper wasn't accepted for publication until 1959. Also in 1959,  an unrefereed  note on the special case of a moving sphere was published by Roger Penrose shortly after his doctoral dissertation.

(2023-08-09)   .
What happens when the occupants of a lone spaceship feel a constant acceleration g?.

(2023-08-05)   .
The irreducible flaw in Special Relativity.

One would try in vain to explain what it is that one should understand by
thepure and simple acceleration of a body. [...]One would succeed only
in defining the relative accelerations of bodies with respect to each other.
Albert Einstein (1914)

In 1914, one year before he'd publish hisGeneral Theory of Relativity, Einstein critiqued at length the internal consistency of the Special Theory of Relativity. The Special Theory is explicitely limited to uniform motion in inertial frames. Yet, there's no way to check that a frame is precisely inertial, without first calibrating itin a frame assumed to be inertial.  An inertial observer has to maintain a constantspeed and direction relative to the rest of the Universe, which is a non-local definition.

What is the transformation law for accelerations

For simplicity, let's consider only the one-dimensional case for three corpusculesat the origin:  O, A and B. If a  is the acceleration of  A  relative to  O  and b  is the acceleration of  B relative to  O,  what is the acceleration of  B  relative to  A? No,  it's not b - a ...