[45064] in Discussion of MIT-community interests
Take a look at your record, the neighbor, your spouse
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Mon Jun 22 11:24:38 2015
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<h1>We have thus analyzed an event-the collision of the light ray with Or---from<br />
two different points of view. G. says that for this event,<br />
x’ = 0 and A+’ z 2L’<br />
C<br />
R. says that for this event,<br />
x = “Af = -P’/c)<br />
(1 - vz/c 2) I,?<br />
and<br />
(i!L’,ic)<br />
Af = --__<br />
(1 - vZ’/c ) 2 I,?<br />
(3.18)<br />
(3.19)<br />
(3.20)<br />
There are several things ‘we can do with this information. For example, the ratio<br />
of At to At’ can be obtained. Division of the equation for A.t by that for At’<br />
3.17 The dilation 63<br />
gives At/At’ = (1 - v’/c’)-‘~!, or<br />
Af = Atr<br />
( 1 - ?/c2)1’2<br />
(X21)<br />
That is, the observers obtain different times for the occurrence of the event.<br />
T o g i v e a n u m e r i c a l e x a m p l e , i f Y = 4c/5, 1 - - v2/c2 = ‘/,s a n d A t =<br />
At1/(9/25)‘/2 = (%)At’. So if G.‘s clock, at 0’, reads 3 set elapsed time, then a<br />
clock at rest in R.‘s system, which is at the position of 0’ when the ray strikes it,<br />
will have beat 5 seconds.<br />
Hence the “moving” clock, G.‘s clock, beats more slowly than R.‘s clocks. In<br />
this experiment G.‘s clock at 10’ was the only clock he used in making the measurements.<br />
However, R. useed one clock at his origin to mark the time the ray<br />
went out, and one clock at the final position to mark the time of arrival of the<br />
ray back at 0’. These two clocks in R.‘s system cannot be the same clock because<br />
we assumed in the tllought experiment that all R.‘s clocks remain at rest in R.<br />
R., therefore, used at least two clocks. We may conclude that for the speesd of<br />
light to have the same value for all observers, it must be true that clo&s moving<br />
relative to a system beat slower than clocks at rest in the system. However, the<br />
observer at rest must use at least two clocks to see the effect, while the moving<br />
observer carries one clock along with him. This effect is called time dilatron.<br />
In this experiment,, the clock carried by the “moving” observer, G., appears<br />
to beat more slowly than the two clocks in the “rest” system, that 01: R. If we<br />
analyze a similar experimerit from the point of view of G., in which we regard<br />
G.‘s system as the rest system, then by the principle of relativity we must find<br />
that a single clock carried along by R. will beat slower than G.‘s clocks. In this<br />
latter experiment, by the principle of relativity,<br />
j ,<br />
A t ’ = At<br />
F-<br />
,-!I!<br />
‘\<br />
)<br />
C2<br />
(3.22)<br />
,. _. _.<br />
just the opposite of Equation (3;27). This shows that the student should not<br />
attempt to learn the equations of relativity in terms of primed and unprimed<br />
variables, but in terms of the physical interpretation of the equations; confusion<br />
regarding the sense of the various contraction and dilation effects is then less<br />
likely to result.<br />
Suppose someone your own age gets in a rocket ship and moves past you with a<br />
1 ‘I2<br />
speed Y such that (1 - v2/c ) = % Suppose that in 10 seconds, by his own<br />
reckoning, he counts that his heart beats 10 times. You would observe that in ten<br />
seconds, by your own reckoning, his clocks have recorded less than ten seconds,<br />
or (% )( 10) = 5 set, so you would observe that his heart beats only tii times. If<br />
he goes to Mars ancl returns, he will then be younger than you when he gets<br />
back.<br />
This time dilotion effect has been observed in experiments in which the average<br />
lifetimes of high speed particles called p-mesons are measured. p-mesons at rest<br />
decay into electrons after arl average lifetime of 2.2 x lo-* sec. This decay<br />
can be thought of as an internol clock in the meson. When the mesons are moving<br />
64 Special theory of relativity<br />
rapidly, it appears to us that the internal clock beats slower, so the meson’s<br />
average lifetime appears to be longer. Table 3.3 gives some experimentally<br />
observed lifetimes, together with correspolnding values calculated from Equation<br />
(3.21) for differing values of v/c. The measurements were obtained by B.<br />
Rossi and D. B. Hall, who counted the number of cosmic ray /J mesons at different<br />
heights above the ealrth’s surface. Since the p’s are produced by high energy<br />
cosmic ray particles striking the earth’s outer atmosphere, the number of p’s<br />
counted at a given heisght was a measure of the number of p’s surviving after<br />
being created in the primary collisions. From these measurements, along <with<br />
independent measurernents of speeds of the p’s, the lifetimes as a function of<br />
speed could be found.<br />
TABLE 3..3 M-Meson Lifetimes As o Function of Speed<br />
v/c Tmvwtsg. (ohs.) T - - - areroge (Cal4<br />
0.9739<br />
0.9853<br />
0.990<br />
10.6 k 3.5 psec 9.3 psec<br />
13.3 f 5.8 psec 13.0 psec<br />
15.6 k 5.8 psec 15.7 psec<br />
3.18 LENGTH CONTRACTION<br />
The results of the thought experiment, from which we obtained the time dilation<br />
effect, can also be used to derive a length contraction effect for rods oriented<br />
parallel to the relative velocity. Suppose R. has a measuring rod along his x axis,<br />
on which he makes a scratch at his origin 0 and another scratch at the point<br />
ivhere the light ray hits 0’ after reflection from the moving mirror. Let us call<br />
the distance between scratches in R.‘s system Ax. Since Ax is the distance between<br />
0 and 0’ after tile time At, during which G. is moving away with speed v,<br />
Ax = vAt (3.23)<br />
Now the distance, Ax’, measured by G. between the scratches is a distance between<br />
scratches on a rod which is moving with speed v relative to him. It is also<br />
the distance between 0 and 0’, measured after the time, At’, when the light<br />
a f t e r g o i n g f r o m 0 ’ to M’ a r r i v e s b a c k a t 0 ’ . T h e n t h e d i s t a n c e b e
t w e e n<br />
scratches is, for G.,<br />
Ax’ = vAt’ (3.24)<br />
Division of the expression for Ax’ by that for Ax leads to<br />
-Ax’ = -A-t’.<br />
Ax At<br />
(3.25)<br />
Hut from the time dilation ‘equation, Equation (3.21),<br />
Therefore,<br />
Ax’ = _ !f’j’* Ax<br />
c2/<br />
(3.26)<br />
(3.27)<br />
3. I9 lorenfz tronsformarions 6 5<br />
Here Ax is the length of aln object measured in a system in which thtr object is<br />
at Irest. The object is moving with speed v relative to the G system in which the<br />
corresponding lengttl Ax’ is measured. Thus, if an object is moving rlelative to<br />
the observer with velocity v, it appears contracted in the dimension parallel to<br />
v b y t h e f a c t o r , ( 1 - v~/c~)“~. S i n c e y = y ’ a n d z = z ’ , t h e o b j e c t is, n o t
<br />
changed in size in directions perpendicular to v. For example, if v/c =: ys,<br />
Ax’ = (7s ) Ax. This result says that a stick of any length Ax at res,t reloltive<br />
to R., when measurecl by G., appears to be shorter. This effect, in whic:h moving<br />
rods appear contracted in the direction of motion, is a necessary consequenc’e of<br />
the assumption that the speeld of light has the same value for all observers.<br />
Suppose G. and R. both Ihave meter sticks parallel to the x and x’ axes. To<br />
R., the length of G.‘s stick appears to be less than a meter. Also to G., R.‘s stick is<br />
less than a meter lonlg. How can each measure the other’s stick to be! shorter?<br />
The reason is that to measure a moving length one must find the positions of the<br />
two ends simultaneously, and then measure the distance between these positions.<br />
The two observers simply disagree about what measurements are simultaneous,<br />
as we shall see. It should be noted that if the physical situation is, reversed<br />
so that the length is at rest relative to G., Equation (3.27) woul’d become<br />
Ax = (1 - v’/c’)” Ax’. So, as in the case of time dilation, one should not<br />
learn the equation in terms of where the prime goes but iln terms of the physical<br />
situation corresponding to the given equation.<br />
-19 LORENTZ TRANSFORMATIIONS<br />
With the information gainecl from these thought experiments, we can now find<br />
the Lorentz transformations which give the relativistic relations between coomrclinotes<br />
of events, observed frorn different inertial frames. Two of the lequatiolls<br />
a r e u n c h a n g e d : y ’ == y a n d z ’ = z. We will assume in our transformations<br />
that t = f’ = 0 when the origins 0 and 0’ coincide. This can be done by simply<br />
setting the clocks to ziero at that instant.<br />
Suppose an object at P’ in figure 3.15 is at rest relative to G. The distclnce<br />
x i n t h e f i g u r e i s t h e x c o o r d i n a t e o f P’ relative to R; it is the distance<br />
measured parallel to the x oxis, from x = 0 to P’. As me’asured by R.. the distance<br />
from 0’ to P’ is AK = x - vt. To G., the distance O’P’ is simply<br />
J(’ = Ax’. Also, we note that Ax’ is a distance between points at rest in the<br />
moving system G. Thus Ax is less than Ax’ by the factor (1 - v2/c2)‘? We<br />
t h e n h a v e A x ’ = A.</(1 - v’/c’)“~. B u t a s w e f o u n d a b o v e , A x ’ ==
x ’ (and<br />
Ax = x - vt. Therefore, we obtain the following transforlmation equation relating<br />
x’to x and t:<br />
I<br />
x :-<br />
( 1 - vZ/cZ)“2<br />
( x - vt)<br />
This applies if P’ is any point whatever. Hence, if some event occurs relative to<br />
R at position x and at time f, then substitution of x and t into this transforma,tion<br />
equation gives the value of 10 at which G. observes the event. Equation (3.28)<br />
is the same as the ‘corresponding Galilean equation, except for the factor<br />
6 6 Special theory of re/ativ,ify<br />
z<br />
- x, x’<br />
d<br />
Figure 3.15. x coordinate of on object at rest in G and observed by R.<br />
l/( 1 - vz/c2)“‘. As v/c approaches zero, this factor approaches one. Therefore,<br />
the correspondence principle is satisfied.<br />
The principle of relativlity implies that the equation giving x in terms of x’ and<br />
f’ is of the same form as the transformation equation, Equation (3.28), but with<br />
the sign of v reversed. Hence, in terms of x’ and t’, we must have<br />
1<br />
X= - ( x ’ + Vf’)<br />
( 1 - Yz/cz)“2<br />
Finally, we want to find the transformation equation which gives the tirne t’<br />
in terms of measurements made by the observer R. To do this, we use the expression<br />
for x’ of Equation (3.28) to eliminate x’ in Equation (3.29). The resulting<br />
equation is<br />
x = -___1___ 1<br />
(‘I - vZ/c2)“2<br />
[<br />
--~ ( x - vf) + vt’<br />
( 1 - Yz/cz)“z 1 (:3.30)<br />
On solving this last equation for t’, we find that<br />
t’ = 1<br />
272 t - :<br />
(1 - “2/C ) ( ) C<br />
(3.31)<br />
This is the desired relationship giving t’ in terms of t and x. Likewise from the<br />
principle of relativity, expressing f in terms of t’ and x’, we must have<br />
t= 1 - if’ + 5)<br />
(l-v21c) \<br />
2 l/2<br />
(3.32)<br />
For v << c, the two equations, (3.31) and (3.32), both reduce to 1’ = t. Therefore,<br />
the correspondence principle is satisfied.<br />
These equations were found using the length contraction equation. They also<br />
agree with the time dilation formula. We con see this by supposing that a single<br />
clock is at rest in the rnoving system G at x’ = 0. The equation<br />
3.20 !jimultcmeity 6 7<br />
t :I:<br />
1<br />
( 1 - “2/C2)“2( )<br />
t’ + YXI<br />
cz<br />
(3.33)<br />
becomes, for this clock, t = t’/(l - Y’/c’)~‘~. The time t’, read on this single<br />
clock at rest in the moving system G, is therefore less than the time t meas,ured<br />
by a coincident clock in the rest system R. This agrees with Equation (3.13),<br />
found previously from a thousyht experiment.<br />
The set of transformations we have found between x’y’z’t’ and xyzt are:<br />
x’ =: 1<br />
2 2 ,,2 (x - v%<br />
(1 -v/c)<br />
t? q :<br />
1<br />
(, _ v2,c2)l/2 t - y-<br />
( i<br />
(:3.:34)<br />
These are called the Lorentz t,ransformations. We have seen that they satisfy the<br />
correspondence principle. They were derived by repeated use of the two postulates<br />
of the theory oif relativity.<br />
imp/e Suppose that G. is moving away from R. in the positive x direction at a speed<br />
s u c h t h a t v/c =: %,, I f R . sets o f f a f i r e c r a c k e r a t y = z = 0 , x = 1 0 , 0 0 0 m,<br />
t := 10m4 set, where and when does G. observe it?<br />
htion F o r v/c = 5/13, ( 1 - - v2/c2)“” = ‘/,, . Then substitution into Equations (13.34)<br />
gives y’ = z’ = 0, .K’ = - 1 6 6 7 m , t’ = 0 . 9 4 4 x lo-‘!sec.<br />
I.20 S I M U L T A N E I T Y<br />
Aside from the time dilation factor (1 - v~/c~)~“~, the equation for t’ in the<br />
Lorentz transformations differs from the Galilean transformations by a term<br />
Y<br />
Figure 3.16. A number of explosions ot different positions along the x axis are simultoneous<br />
in R.<br />
6 8 .Specia/ theory of relotidy<br />
proportional to x. To see the physical significance of this term, suppose that R.<br />
sets off a number of explosions along the x axis, which by his own clocks occur<br />
s i m u l t a n e o u s l y , a t t h e insstant t = 0 . ( S e e F i g u r e 3 . 1 6 . ) T h e e q u a t i o n t’ =
<br />
(t - vx/c2)/(l - v2/c2)““, t e l l s u s t h a t f o r t = 0 b u t f o r d i f f e r e n t p o s i t i o n s
<br />
x,t’ = -vx/c2(1 - v~/c”)“~. These are then the readings on the various clacks<br />
of G. for the different explosions, all of which are observed simultaneously in<br />
R. at t = 0. Thus, for positive x, these clocks in G appear to be set behind what<br />
R. would call the correct time by the amounts vx/c2(1 - ~‘/c’)“~, which depend<br />
on position. Hence events that appear simultaneous to R. do not appear simultaneous<br />
to G.; the tirnes of their occurrence depend on the x positions of the<br />
events. Simultaneity is thus a concept which has no absolute meaning.<br />
Perhaps we may understand this by considering the observer R. standing, as in<br />
Figure 3.17, halfway between two light detectors /I1 and D2, that record the time<br />
Dl r rl \‘\\A/<br />
Figure 3.17. Light emitted from o point halfway between two detectors at rest in R<br />
arrives simultaneously at the two detectors in R.<br />
at which light hits them. If R. turns on the light bulb he is holding, then, since<br />
it takes the same time for the wavefront to travel from R. to D, as to D2, the<br />
detectors record equal times when light hits them. R. would say that the light<br />
hit the detectors simultaneously.<br />
However, if, as shown in Figure 3.18, G. i,j moving past R.‘s position at the<br />
instant R. turns on the light, then as far as G. is concerned, he sees D, and1 02<br />
rnoving backward wit11 speed Y. Then, in G,.‘s system, the light wave going forward<br />
and the detector D2 are approaching each other, while D, is moving<br />
parallel to the light wave going backward, The light wave going forward ,thus<br />
appears to have traveled less distance before it strikes the detector. Therefsore,<br />
in G.‘s system, the light hits D2 before it hits D,, and the events which were<br />
simultaneous in R are Inot simultaneous in G. Since G. believes that the light hits<br />
D2 first, but D1 and D2 record the same time, G. says that the timer at D2 is set<br />
fast compared to that at D,. That is, the timers are not synchronized in G.<br />
3 . 2 0 .Simu/taneify 6 9<br />
Figure 3.18. Light emitted os G. passes the midpoint between two moving detectors<br />
does not arrive at the detectors simultaneously in G.<br />
,+ 1. Two events at x := +lOO km are observed by R. at the instant t = 0. Whlen<br />
are these events observed by G. if G.‘s velocity in the positive x direction relative<br />
to R has magnitude c/10? (Assume 1’ = t = 0 when x’ = x = 0.)<br />
ution t ’ = -VX/C’(l - V”/C’)“=:’ -0.1(&100)/(3 x 105)(0.‘99)“2<br />
= +3.35 X 10e6 s e c .<br />
mp/e 2. If the relative velocity ha:, magnitude 9c/lO, when are they seen?<br />
~tion t’ := 0.9(*100)/(3 :< 1O5)(O.19)‘/2 = b6.88 X 10e4sec.<br />
When a length melasuremcmt of a moving object is made, the positions of both<br />
ends of the rod must be marked at the same time. (See f’igure 3.19.) Thus, for<br />
R<br />
Figure 3.19. To measure the length of a moving rod, R. makes marks simultaneously at<br />
the positions of the left o nd right ends, XI and xR. L = Lrnarkr.<br />
a r o d o f l e n g t h I a t r e s t i n G , R . c o u l d m e a s u r e i t s a p p a r e n t l e n g t h L b’y
<br />
noting the position of its lef+ end, x r, and the position of its right end, xR, at<br />
the same time, and then measuring the difference xR - XL. Suppose, for example,<br />
70 lipecial theory of relativify<br />
that when G.‘s speed is 10,000 ft/sec, R. waits 10m3 seconds to mark the right<br />
end of the rod after he marks the left end. The error he would make in his length<br />
measurement would be (1 OOOO)( 1 Om3) = 10 ft.<br />
Let R. mark the positions of two ends of the rod at time 1. Then, from the<br />
Lorentz transformations, G. would say the right mark was made at the ,time<br />
1; = (f - VXR/C2)/(1 - “2,/C 2) ‘ I.2 Also, G. would say the left mark was made at<br />
the time t; = (t - vq/c2)/(l - v2/c2~‘2.Since these times are not the same, the<br />
marks do not appear to be made simultaneously in G; rather, it appears the<br />
right end is marked first. The difference between these times is At’ = t; -- t;,<br />
given by<br />
At’ =<br />
V(XR - XI) Vl =<br />
2(1 - “2/c2)‘12 c2(1 - Y2/c2)1’2<br />
(3.35)<br />
In this time, relative to G. the R system moves a distance vAt’ to the left. Hence,<br />
the righthand mark approaches the left end of the rod by a distance<br />
“At’ = --‘2~I.<br />
2(1 - - 2/C’) “2<br />
Thus, if to G. the length of the rod is I’, the distance between the marks is<br />
1 korkr = I’ - -LV2L<br />
c2(1 - - Y2/c2)“2<br />
(3.37)<br />
To R., of course, the distance between the marks is Lma,tr = L, the apparent<br />
length of the rod.<br />
--d-R moves to left before<br />
I xL is marked<br />
II<br />
-Rat instant xI is marked, t:<br />
XR I with t’, > t’,<br />
I<br />
Figure 3.20. R.‘s measurements of the length of the rod in G, as seen by G.<br />
3.2 I Tronsformotion of velocities 7 1<br />
The above result can be used to check the length contraction effect, for suppose<br />
the ratio of the apparent length of a moving rod to that of an identical rod<br />
at rest is denoted by ‘I/y, where y is some constant depending on relative speed.<br />
T h e n s i n c e t o R . t h e r o d a t r e s t i n G i s m o v i n g , Lma,,.. = L = l’/y. Iiowever,<br />
to G. the marks at rest in R are moving with the same speed, so LAark, = I./-y.<br />
Thus, eliminating L’ alld Lk,,k. from Equation (3.37),<br />
1-<br />
:= yl -<br />
21<br />
Y cZ(l - vZ/cZ)“2<br />
(3.313)<br />
T h i s q u a d r a t i c e q u a t i o n f o r *y h a s s o l u t i o n s y = l/(1 - v2/c2)” and :r =<br />
-( 1 - v~/c~)“~. Since the second solution becomes -1 as v/c goes to zlero,<br />
it does not satisfy the correspondence principle and may be discarded. The first<br />
solution agrees with the length contraction found previously from another<br />
thought experiment. Since tke present argument is based on the disagreemelnt<br />
regarding simultanei,ty between the two frames, we see that this is the basic<br />
reason why lengths in one system may appear shortened in another system, ranId<br />
vice-versa.<br />
2 1 TRANSFORMATION OF VIELOCITIES<br />
It is extremely useful to know how velocity measurements made by different<br />
observers are related. Suppose, as is illustrated in Figure 3.21, that R. observes<br />
z<br />
Figure 3.21. The position vector of o particle changes by dr in time dl’.<br />
a particle moving in .time dt from the point with coordinates x,y,z to x + dx,<br />
y + d y , z + d z . I n R t h e v e l o c i t y t h e n h a s c o m p o n e n t s o f dx/dt, dy/dt, land<br />
dz/dt. Suppose G. observes the very same particle going from x’, y’, z’ to<br />
x’ -t dx’, y’ + dy’, .z’ + dir’ in the time interval dt’. The velocity
colnponents<br />
in G are then dx’/dt’, dy’/dt’, and dz’/dt’. We shall use the Lorentz trclnsformation
<br />
equations to find dx’, dy’, dz’, and dt’ in terms of the unprimed differential<br />
quantities. The use of the velocity definitions just stated will then lead to the<br />
velocity transformations.<br />
7 2 Special theory of relofivify<br />
One of the equations of the Lorentz transformation, Equations (3.34), is<br />
x’ zz 1<br />
2 ,T2 (x - 4<br />
(1 - v2/c )<br />
the differential form of this equation is<br />
d x ’ =<br />
1<br />
2 1/2 (dx - VW</h1>
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