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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Theory of Relativity

 Albert Einstein 
1879-1955
The great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature.
Albert Einstein (1916)

Related articles:

Related Links (Outside this Site)

Einstein Light  by  Joe Wolfe  &  George Hatsidimitris  (UNSW, Sydney)
Relativity on the World Wide Web  by  Chris Hillman
Reflections on Relativity  by  Kevin S. Brown  (600-page e-book)
General Relativity by  David M. Harrison  (University of Toronto)
The Speed of Light by  R.F. Egerton  (University of Alberta)
Additivity, rapidity, relativity  by  Jean-Marc Lévy-Leblond  &  Jean-Pierre Provost.
Why we believe in Special Relativity by  John S. Reid  (University of Aberdeen)
 
Humor:  Relativity, in Words of Four Letters or Less  by  Brian Raiter
 
Wikipedia :   The Speed of Light   |   Faster than Light (FTL)       DMOZ: Special Relativity

Videos

Simultaneity  |  Time Travel  |  Time Dilation  |  Elegant Universe - Spacetime
(Special)  Relativity in 5 minutes by  Gal Barak.
The Lorentz Transformation (MU42)  by  David L. Goodstein   1 | 2 | 3
Relativistic Field Theory  by  Leonard Susskind   [14 hours]   iTunes | YouTube
 
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 World Year of Physics 2005

Special Relativity

Note about Notations :

The timelike-coordinate of a  quadrivector  is listed  first.  Vectorial quantities are  boldfaced; lowercase symbols are used for 3D vectors, capitalization for 4-vectors:

R  =  (ct,x,y,z)  =  (ct,r)             X = (x0 , x1 , x2 , x3 ) = (x0 , x)

The sign of a 4D dot product is defined to generalize the 3D-space concept:

U . V   =   (u0 , u) . (v0 , v)   =   u . v - u0 v0   =   - u0 v0 + u1 v1 + u2 v2 + u3 v3


(2003-11-03)   Observers in Motion:  The Lorentz Transform 
How are the coordinates in two uniformly moving systems related?

In the framework of the Special Theory of Relativity, such coordinates are linearly related.  Nonlinear relations are the subject of  General Relativity Theory,  where  linear  transforms only apply to  infinitesimal  coordinates (cdt,dx,dy,dz).

Call t,x,y,z the coordinates in one system (S) and t',x',y',z' the coordinates in the other (S').  Assume the axes are so oriented that motion is along the x-axis of S which is also the x'-axis of S'.  For points on that invariant axis, y,z,y',z' are zero, and we have to find proper dimensionless coefficients aij in the relations:

ct'  =  a00 ct  +  a01 x
x'   =  a10 ct  +  a11 x

In this, c is the speed of light in a vacuum (now best called Einstein's constant).  The symbol  c  stands for  celerity  (Latin  celeritas).  Speed and phase celerity are identical for things that propagate at celerity c.

The main tenet of Relativity requires the  important physical constant  c  to be the same for all observers...  That is, (x±ct) is zero if and only if (x'±ct') is.  Now, we have the linear relations:

x'+ct'   =   (a10+ a00 ) ct  +  (a11+ a01 ) x
x'-ct'   =   (a10- a00 ) ct  +  (a11- a01 ) x

The aforementioned conditions can only be met if the coefficients of x and ct on the right hand side of each equation are respectively proportional to the coefficients of x' and ct' on the left hand side.  Therefore:

a10+ a00   =   a11+ a01
a10- a00   =   a01- a11

Adding or subtracting these two equations yields:   a00 = a11   and   a01 = a10
Introducing  g = a00   and letting  b  be equal to  -a10 / a00,  we thus obtain...

The Lorentz Transform:

ct'  =  g ( ct  -  bx )
x'  =  g (-bct  +  x )

The origin of S'  (x' = 0)  moves at a speed v in S  (x = vt).  Therefore:

b   =   v/c

Now, we have to assume that the inverse transform from S' to S is the same as the direct transform from S to S', with only a trivial change of sign (or else, there would be a directional preference in the Universe, which we find repugnant):

ct  =  g ( ct'  +  bx' )
x  =  g ( bct'  +  x' )

Using the prior values of ct' and x', we obtain  ct  =  g2 (1-b2 ) ct.  So:

 
g   =    
 
 1 
vinculum
space
vinculum
Ö 1-b2

FitzGerald-Lorentz contraction of a moving object:

If points at rest in S' are observed at constant time  t = 0  in S,  then we have:

x'  =  g ( 0  +  x )

It is customary to use nought subscripts  ( So = S',  xo = x', etc.)  for the rest frame of a solid  (that's a special case of the  proper frame  of an extended object, a key concept in relativistic thermodynamics).  Therefore, the above relation  x = x'/g  also reads:

vinculum
x   =   xo   Ö 1-b2

Thus, a yardstick moving lengthwise appears shorter than a yardstick at rest because a stationary observer who records both extremities  simultaneously  actually records them at what would be different times for a moving observer traveling with the ruler.

 George Francis FitzGerald 
 1851-1901
George Francis FitzGerald
 Signature of
 George Francis FitzGerald
This shortening of distances, called the  FitzGerald-Lorentz contraction,  occurs only in the direction of the velocity of a moving object  (distances perpendicular to the velocity are not affected).  It was proposed in 1889  (well before  Special Relativity  was formulated)  by the Irish physicist George FitzGerald (1851-1901) to explain the negative result of the Michelson-Morley experiment of 1887.  FitzGerald is also remembered for his prediction that oscillating currents ought to generate radio waves (1883).

A trivial consequence of the  FitzGerald-Lorentz contraction  is that a container has a lesser volume  V  when it's moving than when it's not.

vinculum
V   =   Vo   Ö 1-b2

Other thermodynamical characteristics of extended bodies  (including temperature)  may also depend on their motion, because  simultaneity is relative  (by definition, all parts of a moving body are observed at constant time in the frame of the observer).  The same remark applies to pointlike objects, which can be viewed as  tiny  extended bodies.

Why is time slower in rockets?  by  Pr. Mike Merrifield  ("Sixty Symbols" filmed by  Brady Haran, 2013).


(2005-02-13)   Combining Collinear Velocities
How do relativistic speeds add up? 

Considering the case of motions along the x-axis only, the problem is to find the velocity  w  with respect to S of something moving at velocity  u  in a system S' moving itself at velocity  v  with respect to S.

Well, if  x' = ut',  the above Lorentz transform  (with b = v/c)  tells us that:

x - vt   =   u ( t - xv/c2 )

Solving for x, we obtain   x = wt,  where w is given by the following expression:

 
w   =    
 
 u + v 
vinculum
1  +  uv / c2

Note that, if u and v are both  subluminal  (between -c and c)  then so is w...

However, we need only assume that  one  of the speeds is  subluminal  As phrased above, v is so restricted (it's the speed of one coordinate system in the other) whereas u could well be a  superluminal  phase celerity, for example.  In that case, the result w is superluminal as well.

Nothing is subluminal for one observer and superluminal for another, provided all observers have subluminal motion with respect to each other.

Rapidity :

It can be convenient to introduce a function of the speed  v  called  rapidity, which is essentially the  inverse  of the hyperbolic tangent function.  Namely :

j   =   c tanh-1 (v/c)   =   c th-1 (v/c)   =   c Argth (v/c)

v   =   c  th (j/c)

There's a simple addition formula for the  direct  function  (tanh or th):

th (x+y)   =      th(x) + th(y) 
vinculum
1  +  th(x) th(y)

This parallels nicely the above formula for the addition of  collinear  velocities and shows that  rapidities are additive :  If an object moves at rapidity  j2  relative to a frame of reference of rapidity  j1 ,  then the rapidity of that object is  j1+j2

The concept of  rapidity  was introduced in 1910 by the British mathematician Edmund T. Whittaker (1873-1956) in the midst of an  historical  account of the development of electromagnetism up to and including  special relativity  (which Whittaker attributes to the efforts of Lorentz and Poincaré, rather than Einstein).

A History of the Theories of Aether and Electricity   by  E.T. Whittaker  (1910)


(2011-02-19)   Combining Non-collinear Velocities

Let's generalize the above to velocities along different directions.  Using the notations of the previous section, we now consider the following motion in the frame of reference attached to an actual particle moving at speed  v < c. 

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

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

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

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

y   =   wy t   =   g (wx - v) (tg q) t

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

Cartesian Components of the Observed Resultant Velocity
   
wx   =    
 
 v + u cos q     
wy   =    
 
u (1 - v2/c2 )½ sin q  
vinculum vinculum
1  +  (uv cos q) / c2 1  +  (uv cos q) / c2

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

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

The reader may want to retrieve the above addition formula for parallel velocities by putting  cos q = 1  in this lesser-known general relation.

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

Relativistic Aberration of Light
  wx  
   =   cos a   =    
 
 (v/c) + cos q   
vinculum vinculum
c 1  +  (v/c) cos q

Wikipedia:   Relativistic Aberration


Kreuzer   (Yahoo! 2011-02-12)   The Headlight Effect
Isotropic radiation is focused forward if the source moves.

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

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

wx   =   v         and         wy   =   c / g

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

Wikipedia:   Relativistic Beaming (Searchlight Effect)
Video:   Optical Effects of Special Relativity


(2010-12-24)   The Lord is Subtle, not Malicious.
On the  straight  addition of closing speeds.

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

The previously discussed  relative speed  of two moving objects is defined as the speed 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 distance between 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  and an 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 the closing speed of two objects  (because two moving observers are supposedly experiencing the same flow of time).  In relativistic mechanics, this ain't so.

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


(2005-05-05)   Propagation of Light in Moving Bodies   (Fresnel, 1818)
Dependence of the  Fresnel drag  f  on the refractive index  n  (Fizeau).

 Fizeau
 experiment

In 1851, Hippolyte Fizeau (1819-1896) used interferometry to determine how the speed of a moving liquid affects the propagation of light in it.  He found an empirical relation between the magnitude of the effect and the refractive index (n) which was later explained relativistically...

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

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

w = (u + v) / (1 + uv/c2 )
= (c/n + v) / (1 + v/nc) 
= c/n   +   v ( 1-1/n2 )  [ 1 + v/nc ] -1
» c/n   +   v ( 1-1/n2 )
» c/n   +   v f

The parameter   f =  1-1/n2   is known as the  Fresnel drag coefficient.  It has been named after  Augustin Fresnel (1788-1827)  who introduced the now obsolete aether drag hypothesis in 1818, to explain experiments performed by Arago 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 partial dragging.

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 principle of relativity  (Lorentz did it)  Einstein considered it an excellent experimental test of  Special Relativity.


(2005-07-27)     Harress-Sagnac Effect  (using mirrors or fiber optics)
Rotation rate of an optical loop is revealed by interfering opposite beams.
   
 Sagnac's Rotating
 Interferometer

The  Sagnac effect  is mainly a  nonrelativistic  effect observed when two coherent light beams  (which may come from a single source through a half-silvered mirror)  travel in opposite directions around an optical loop.  When the apparatus rotates, a phase difference is observed which is essentially proportional to both the  pulsatance  w  [the rotation rate in rad/s]  and the area of the loop  (actually, the  apparent  area, for a distant observer located on the axis of rotation).

The thing  shouldn't  be interpreted as a change in the celerity of light, supposedly due to rotation  (dwelling on this fallacious point is a misguided endeavor which is popular in the pseudoscientific community).  It's mainly that the beams must travel different distances to their rendezvous, at a point which will have moved toward one beam and away from the other, by the time they get there !

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).

In some of the so-called "laser gyroscopes"  (devised around 1963)  of modern guidance systems, the latter method is now used to produce a resonant frequency proportional to the rate of rotation.  One advantage over mirrors is that a fiber optic cable can easily be coiled to increase the effective area of the loop within a compact volume.  Rotation rates around  5 ´ 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 with many 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 point of the loop travel a distance  2pR  in a time  2p/w.  Therefore, the nonrelativistic expression for the speed  v  of each point of the loop does hold:   v = wR.  (The FitzGerald-Lorentz contraction applies to the length of moving objects, not to the space they travel.)

Using this value of  v  in the (exact) expression from the previous article, we obtain the value 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   +   wR ( 1-1/n2 )  [ 1 ± wR/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  wt  when the beam reaches it again after one turn, so that  t  is a solution of:

t w ±  / R    =    wt ± 2p

Solving for  t,  we get:    t-  =  2pR / ( -w- + wR )       t+  =  2pR / ( w+ - wR )
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-     =     Dt     =       4p R 2 w
vinculum
c 2 - w 2 R 2

With ordinary objects, the above is undistinguishable from  4p R 2 w / c2  and is thus proportional to the rotation  w.  Indeed, the speed  wR  of the circumference is  much  lower than the speed of light  (the relative error is about  1.22 ´10-13  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 by Oliver Lodge (1851-1940) in 1893 and by Albert A. Michelson in 1904.

In 1911, Francis Harress tried to substitute glass for the liquids used in Fizeau's investigations.  He had the idea to observe rotating rings of glass, but could clearly do so only at fixed points of such rings...  As we've discovered theoretically in the above discussion, the resulting effect does  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 own experimental 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 rotation of the Earth, with a rectangular optical loop  0.2 mile wide and 0.4 mile long.

Earlier studies of the Sagnac effect  by  Grigorii B. Malykin  (1997)
Sagnac Effect   |   Ring Laser Project   |   Wikipedia


gerry (2002-06-27)   Relativistic Limit of a Relative Speed
What is   w = (u-v) / (1-uv/c2 )   as both u and v approach c ?

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 u and another train (V) moves at speed v (both speeds being measured relative to some platform on which the "observer" is located).  According to the  Special Theory of Relativity  (see above)  the quantity  w = (u-v) / (1-uv/c2 )  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 moving parallel 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 rocket might be moving...

To make this relativistic math more transparent, you may want to consider rapidity instead of speed.  The interesting thing is that rapidities are additive, whereas speeds are not:  The rapidities x, y, z corresponding to the above speeds u, v, w are thus related by the 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...


gerry (2002-06-30)     [follow-up to the previous article] 
What's the relative velocity of two photons?

The velocity of object B relative to object A is the velocity of B measured in the frame of reference 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 definition can be consistently extended to include objects moving at speed c.  Such an extension exists, except in the very special case of two photons moving in exactly the same direction, where the notion of "relative speed" breaks down completely (as shown in the above discussion of the relativistic formula for the "addition" of parallel velocities).

If A is a photon but B is not also moving at speed c, we may still reach a firm conclusion by noticing that the velocity of B relative to A must be the opposite of the velocity of A relative to B (which is well-defined, unless B moves at speed c).

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

c (u-v) / ||u-v||

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

On the other hand, if u and v are equal, we are back to our previous discussion:  Two sub-c chasers with any arbitrary relative velocity (not necessarily collinear with u = v) could both approach the velocity (u = v) of both photons.  Therefore, there is no continuous way to define the relative velocities of two photons moving in the same direction.  Having said this, we may or may not find it useful to state (rather arbitrarily) that two such photons have zero relative speed.  However, we failed to find a compelling reason for this "obvious" choice and cannot guarantee that it would not lead to a paradox of some kind...

 Hermann Minkowski 
  1864-1909
Hermann Minkowski

(2005-04-14)   Minkowski Space and 4-Vectors
4D objects transforming according to the Lorentz rule

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)

Hermann Minkowski (1864-1909)  spoke those words on September 21, 1908, in the opening lecture  (entitled  Raum und Zeit  or  Space and Time)  for the mathematical section at the annual meeting of the  German Association of Natural Scientists and Physicians  (GDNÄ = Gesellschaft Deutscher Naturforscher und Ärzte) in Cologne, Germany.  Weeks later  (on January 12, 1909)  Hermann Minkowski would die of a ruptured appendix, at the age of 44.

Minkowski was explaining that a timelike component is associated to any physical 3-vector which makes the [contravariant] coordinates of the resulting 4-dimensional object transform according to the same Lorenz transforms which applies to the 4-dimensional interval between two pointlike events.

 Come back later, we're
 still working on this one...

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

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

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


(2006-03-28)   Three-Dimensional Expression of the Lorentz Transform
A Lorentz  boost  of speed   V = cb   may be expressed vectorially.

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

(A.b /b ) b /b         and         A  -  (A.b /b ) b /b

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

a' =  g  ( a  -  A.b )
A'= A  +  [ (g-1) A.b /b2  -  gb

linear  transformation which preserve spacetime intervals  (while respecting the orientation of space and the direction of time)  is necessarily a composition of such a  boost  with a spatial rotation  (which leaves time unchanged and preserves spatial distances without changing the orientation of space).  Such transformations form the  6-dimensional  Restricted Lorentz Group  (a spatial rotation may be specified by its 3  Euler angles, while a boost may be given by the 3 components of the vector  b  introduced above).

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


(2005-04-16)   Wave and Phase:  Wave Vector = Phase Gradient
An Introduction to Four-Dimensional Wave Vectors.

The  phase  j  of a periodic phenomenum describes its 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:  j  is a relativistic invariant.

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

j   =   wt - k.r

The pulsatance  (w)  is proportional to the wave's frequency  (n),  whereas the magnitude of the 3D wave-vector  (k)  is tied to the wavelength  (l), namely:

w= 2p n
|| k ||= 2p / l

The celerity of the wave  (its phase speed)  is the product  u = ln.

K = (w/c, k)  is a 4-vector because it's the 4D gradient of a scalar invariant:

K   =   (w/c , k)   =   Grad(-j)

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)   Doppler Shift.  Relativistic Transverse Doppler Effect 
The radial effect is multiplied by an isotropic relativistic factor.

Among redshift factors, the classical  Doppler effect  comes from a changing distance between the observer 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  w0 = 2pn0  (in its own rest frame)  is given by the Lorentz transform for the wave-vector:

w0 /c  =  g ( w/c  -  b k x )  =  g ( w/c  -  k.v/c )

 Doppler Effect

Therefore, calling  q  the angle between v and the direction of observation (-k) we have:

n0  =  g ( n  +  cos(q)  ||v|| / l )

If the signal propagates at  celerity  u = ln  (not necessarily c)  in the frame of the observer, we obtain the following relation  (see note below if  u<c):

Vinculum
    l n0  Ö  1 - v2/c2     =     u + v cos q    

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

In this context, that relativistic correction factor is called the "Transverse Doppler Effect".  That discovery is universally attributed to Einstein, even by authors who prefer to credit other foundations of Special Relativity to Poincaré or Lorentz...  Einstein used it in September 1905 to establish the inertia of energy  (E = mc).

The above is of the form   l n0' = u' :   The classical radial celerity (u') is the wavelength (l) multiplied into a relativistically adjusted frequency (n0').

A Dubious Academic Tradition:

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

Relativistic Doppler effect for light  (u = c):  Values of   1 + z  =  l / l 0
Outbound  (q = 0) Transverse  (q = ±p/2) Inbound  (q = p)
[ (1+v/c) / (1-v/c) ] ½ 1 / (1-v2/c2 ) ½ [ (1-v/c) / (1+v/c) ] ½

Note, when  u < c :

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

For light in a vacuum, this is of no concern because the main tenet of special relativity does state that something that propagates as a wave of celerity  c  in one particular inertial 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 of the propagation medium  (e.g.,  air in the case of sound).  Otherwise, we are faced with a 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 same velocity with respect to the propagation medium, as when one listens to an outdoor concert in a steady wind).


(2005-04-14)   Energy (E) and Momentum (P)
E/c and P form a 4-vector  (i.e., they transform like ct and r).

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

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

One approach of Newtonian mechanics which can be consistently generalized to the framework of  Special Relativity  is to introduce the notion of  linear momentum  (the product  P  of mass  m  by velocity  v)  and to postulate that the time derivative of that quantity is a vectorial quantity, called  force, which somehow describes dynamical exchanges 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 a fancy way of saying that every variation in the momentum of one part will always be exactly compensated by the variation in the momentum of another, so that  total momentum is conserved.

 Come back later, we're
 still working on this one...

Einstein's derivation of the inertia of energy  by  Terence Tao.
Einstein's proof of  E = m c2  by  Henry Reich   (Minute Physics)


Louis Vlemincq (Belgium; e-mail 2005-04-14)   Kinetic Energy
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 g)  c 2   =     m c 2
vinculum
space
vinculum
Ö 1 - v2/c2
Unfortunately, this is also written  E = m c2  by letting the symbol  m  be the so-called  relativistic mass,  which we denote here by  m g  as we reserve the symbol  m  itself for the so-called "invariant mass" or "rest mass" of a particle  (following current standard practice).

Under the assumption that v is much smaller that c, a good approximation for E is 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, whereas the second term is called  kinetic energy.  Actually, the first term was originally conjectured by Einstein because of the above considerations:  It's just mathematically simpler to assume that a motionless body of mass m already has an energy  mc2.  This explains painlessly the so-called  mass defect  observed in the decay of radioactive elements :  A nuclear decay always leaves remnants whose rest masses add up to less than the mass of the original nucleus.  The balance of the energy appears in the form of either kinetic energy or radiation...

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

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


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

Particles travelling at speed c can only have a finite energy if they have zero rest 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 associated wave:  The energy of a photon of frequency  n  is  hn, where h is Planck's constant.  This relation was proposed by Einstein (1905) in his explanation of the laws of the photoelectric effect  (for which he received the Nobel Prize in 1921)  which may be construed as a formal  discovery  of the photon...

 Arms of Isaac Newton 
 1643-1727 Arms of Christiaan Huygens 
 1629-1695  Arms of James Clerk Maxwell 
 1831-1879
At first, Isaac Newton (1643-1727) had argued that light was  corpuscular  [i.e., consisting of discrete particles]  against  Christiaan Huygens (1629-1695) who held the view that it was  undulatory [wavelike].  The viewpoint of the latter was revived in 1803, when Thomas Young's double-slit experiment proved light was capable of interference.  The same doctrine was reinforced when James Clerk Maxwell (1831-1879) put forth his famous differential equations governing electromagnetism in general, and light in particular [at a macroscopic level].  Max Planck (1858-1947) also believed strictly in the wave nature of light, even after his own success in explaining the blackbody spectrum by assuming matter and radiation could only exchange energy in packets ("quanta") proportional to the radiation's frequency.  Thus, it was a revolutionary proposal of Einstein's that Planck's packets of energy could actually be related to the  particles of radiation  envisioned by Newton.

A generalization to electrons and other  subluminal  particles was proposed by the French physicist Louis de Broglie, who put forth a new principle establishing the dual corpuscular and undulatory nature of  everything, as discussed next.


 Arms of Louis de Broglie 
 1892-1987(2005-04-16)   The Principle of  de Broglie  is Relativistic
Corpuscular and undulatory duality, as proposed in 1923/1924. 

In 1923, the French physicist Louis de Broglie (1892-1987; Nobel 1929) was still a graduate student at the Sorbonne when he proposed the idea of matter waves, which he defended successfully in 1924  (with the support of Einstein himself)  in front of a doctoral committee which included Paul Langevin (1872-1946).  At the time, de Broglie stated that his proposed matter waves might be observable in experiments involving crystal diffraction with electrons.  Such experimental confirmations came in 1927, with two independent experiments:  one by Clinton J. Davisson (1881-1958; Nobel 1937) and Lester H. Germer (1896-1971), the other by G.P. Thomson (1892-1975; Nobel 1937)...  Ironically, George Paget Thomson thus demonstrated the undulatory nature of electrons, whose corpuscular properties were established three decades earlier by his own father, J.J. Thomson (1856-1940; Nobel 1906).

De Broglie's idea was that any particle is associated with a so-called pilot wave:  The momentum of one and the wave-vector of the other are  proportional  and the coefficient of proportionality is a universal constant.  We'll state de Broglie's principle using the 4-dimensional wave-vector introduced above:

K= (w/c , k)   =   Grad(-j)
w= 2p n
|| k ||= 2p / l

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-vector K proportional to P, namely:

P   =    h-bar K   =   (h/2p) K       [ h is Planck'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 n
p=h / l

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

The above relations make the celerity  u = ln  equal to E/p.  For a particle of rest 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 between celerity and speed:

    u v   =   c 2    

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

Formerly, the above mechanical speed (v) was identified with the "group speed" which measures the propagation of a wave's modulation  (i.e., the overall shape of its envelope).  Actually, this "fact" was used by Erwin Schrödinger himself in one particular justification for Schrödinger's equation.  However, the mathematical expression for group speed does not necessarily imply such a relation, which is not absolute:  Although it's usually slower than light, group speed can indeed be faster than light.  This was demonstrated experimentally in the 1980's.

De Broglie wavelength of a relativistic particle :

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

l   =     h c
Vinculum
Ö Vinculum
(E - mc2 ) (E + mc2 )

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

l   =     h c
Vinculum
W   Ö Vinculum
1 + 2x

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

vinculum
l   =   h / p   =   (h / mv)   Ö 1 - (v/c)2

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

    l   =   (h/mc)  [ (E/mc2 ) 2 - 1 ] -½    

The de Broglie wavelength of a  240 Mev proton  is  1.74 fm   (Yahoo! Answers)


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

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

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 out in the (x,y) plane, at an angle  q  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 photon from the sum of the two initial  4-momenta.

The Minkowski square of the electron's 4-momentum is always  -(mc)2.  So:

- m 2 c 4   =   -(mc2 + hn - hn' ) 2  +  ( hn - hn' cos q ) 2  +  ( hn' sin q ) 2

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

    n'   =   n  /  [ 1 + (1 - cos q ) n / nc ]    

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

lc = h / mc = 0.0024263102389(16) nm

    l'   =   l  [ 1 +  (1 - cos q ) lc / l ]    

l' - l   =   lc (1 - cos q )

The  Compton shift  (z) is best defined, like other types of  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 q ) n / nc   =   (1 - cos q ) E / mc2    

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

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

With energetic photons  (gamma rays)  the target electrons may recoil at high speed and cause bluish cherenkov radiation in transparent bodies.


(2008-01-31)   Klein-Nishina formula for Compton scattering  (1929)
Differential cross-section of electrons in Compton diffusion.

Photons deflected by an angle between q and q+dq span a solid angle  dW = 2p sin q dq.  Such a deflection occurs with the same probability as would a collision of a classical point particle with an obstacle of  cross section  ds.  For Compton 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) :

ds    =    ½ (re ) 2  [ P  -  P2 sin2 q  +  P3 ]   dW
where   P   =   n'/n   =   [ 1 + (1 - cos q ) n / nc ] -1

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

re   =     q2     =   2.8179402894(58) fm
vinculum
4peo mc2

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

s  =   p (re ) 2  ò 1
-1
  [  (1+(1-x)u)-1 - (1-x2 ) (1+(1-x)u)-2 + (1+(1-x)u)-3  ]  dx
 
=   p (re ) 2   [   Log (1+2u)   -  2   (u+1) Log (1+2u) - 2u   +  2   u+1   ]
Vinculum Vinculum Vinculum
 u  u3 (1+2u)2
 

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

so   =    8/3  p (re ) 2

At high energies, the cross-section  (svanishes  but the average transfer of energy increases logarithmically with the energy of the incoming photon.

Klein-Nishina formula (1929)


Bert Dobbelaere (2008-01-29; e-mail)   Quantum Suppression
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's necessary 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 transfer something like 1/100000 of their energy to the electron they collide with, according to the above formula  (the maximum occurs when the photon bounces straight back to the direction where it came from).

The electrons bound in the ordinary orbitals of chemical elements can 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 by a lone electron.

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


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

dE   =   v.dp   (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:

dE = E'1 - E1 = - ( E'2 - E2 ) = y c
dp = p'1 - p1 = - ( p'2 - p2 ) = x u     [ with  ||u|| = 1 ]

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

pi   =   u . pi   =   (Ei / c 2 ) vi

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

E'12 / c2 - p'12 = E12 / c2 - p12
E'22 / c2 - p'22 = E22 / c2 - p22

In the form   (p'i - pi ).(p'i + pi )  =  (E'i - Ei ) (E'i + Ei ) / 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 following linear system:

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

By itself, the first equation  already  says that   dE   =   v.dp  (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:

 Come back later, we're
 still working on this one...


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

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

However, there are high-energy quantum processes whose net result is similar to an  elastic collision between two photons  (cf. above Feynman diagram)  and we can determine the relations between the incoming and outgoing photons.

If the photons are traveling in  exactly  the same direction, they will never collide  (just as cars which travel at the 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-momentum of what can be called the  center of mass  of those two photons.  The mass of that point is the Minkowski-length of the momentum-energy divided by c.  Its  3D-velocity is the 3D-momentum divided by that mass.

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

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

The azimuthal orientation of the outgoing direction in which the incoming symmetry is broken will also be important to whoever might have to use the relevant Lorentz transform to translate the above simplicity into actual laboratory measurements.

For the head-on collision of two photons of frequencies n0 and n1, we may write the conservation of energy-momentum  (divided by h/c)  as follows:

bracket
bracket
bracket
n0
 n0 
 0 
0
bracket
bracket
bracket
+ bracket
bracket
bracket
n1
 -n1 
 0 
0
bracket
bracket
bracket
= bracket
bracket
bracket
n0'
 n0'  cos q0 
 n0'  sin q0 
0
bracket
bracket
bracket
+ bracket
bracket
bracket
n1'
 -n1'  cos q1 
 -n1'  sin q1 
0
bracket
bracket
bracket

Two-Photon Physics  by I.F. Ginzburg (Acta Physica Polonica, 2006)
Photon-Photon Scattering (Physics Forums)


 Cherenkov glow in 
 a nuclear reactor  
(2005-05-05)   [ Blue ]  Cherenkov Radiation
The Cherenkov effect occurs when a charged particle moves faster than the celerity of light.

In a dispersive medium like a liquid or a gas, the celerity of light depends on frequency.  Normally, the celerity of visible light is below Einstein's constant (c) while the celerity of X-rays is above it.  ( It's the group speed which may never exceed c.  Phase celerity is another matter entirely.)

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

   Lucien Mallet, 1947
Lucien Mallet, 1947
 

However, this is what the  whole  thing was mistaken for until the French radiologist Lucien 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 electrons disloged by Compton 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 and Igor Y. Tamm, following the 1936 discovery of the particular geometry of the Cherenkov beam.  For this, those three men became the first Russians ever to be awarded the Nobel 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  than a 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 :

 Come back later, we're
 still working on this one...

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

Cherenkov Radiation by Philip Gibbs  |  Cherenkov Radiation by Emma Ona Wilhelmi
Cerenkov Light:  What is it?   A Fermilab video.


Alexander   (Yahoo! 2007-08-05)   Constant Acceleration
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 of light, 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 second derivative of his position (x') with respect to time (t') to be constant (g).

 Come back later, we're
 still working on this one...

Wikipedia:   Time Dilation and Space Flight


The relativistic paradox of the twins of Langevin
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 flown around 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.

 Come back later, we're
 still working on this one...

The Häfele-Keating experiment (1971)

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