- Email: [email protected]
VI. Clocks and Time Intervals
VI.
CLOCKS A N D TIME INTERVALS
1. Introduction
The last two chapters are concerned with the topological structure of the class of events (without regard to the length of time intervals). In this chapter, clocks are used to define a time metric, which forms the basis of time geometry. A clock is a physical system that generates and counts a sequence of events at a particle, called the output particle; the output particle of a watch is the tip of its minute hand, and of an atomic clock is the end of its output lead. To measure the time interval between two events a , b by a clock, a, b must occur on the world line of the output particle of the clock, and the number of events generated by the clock between a and b is the value of the interval between them. If no limitation is imposed on the type of clock used, the description of physical phenomena obtained in this way, is at the mercy of the capricious behaviour of the clock: if the period of a pendulum measured by such a clock is found to vary, is the variation due to the pendulum or to the clock itself? Consequently, if a reasonable description of physical phenomena is desired, it is necessary to impose some limitations on the clocks used. In the general theory of relativity, there is a great deal of freedom in the choice of a (coordinate) clock. However, the rates of all clocks are specified relative to coincident standard clocks (by the metric coefficient g,,), and without standard clocks it is not even possible to measure the space-time line element ds. How are we to select a clock? There seem to be two different procedures, both of which are used in practice: (1) Try different clocks, and find out the kind of description of physical phenomena they lead to. The clock that gives the description which is considered by respectable physicists to be the simplest and most natural, is the clock that is crowned to be a standard clock. All other clocks are judged by comparison with it. Until the revolution of the atomic clocks, the king has been the astronomical clock. It has been a 49
50
CLOCKS AND TIME INTERVALS
[Vl 1
benevolent ruler and rendered much service to physics, but alas times do change, and the collective leadership of atomic clocks is taking over. There are very good reasons for this peaceful revolution, which will become apparent before this section is over. (2) Intercompare many clocks, and select the ones that exhibit the least relative instability. Comparison of two clocks means the coincidence of the output particles of the two clocks, and the measurement of the frequency ratio of the number of events generated by one clock to the number of events generated by the other clock between any two end events. The clocks that have the most stable ratio, i.e., the ratio with the smallest amount of fluctuation, are considered to be the clocks with the highest relative stability. This is precisely the procedure used in the selection of atomic time standards. Clearly, the more clocks that agree with each other, the better we feel about their selection. Which type of time standard is better, the solitary type selected by the first procedure or the comparison type selected by the second procedure? There is no getting around the fact that the solitary type standard is arbitrary. It cannot be justified by any theory, because such a theory must involve the concept of a time measure, which depends upon the existence of a standard, which we are trying to justify. The only justification for a solitary type standard is that the description of nature it leads to is acceptable. Unfortunately, what is acceptable at one time may not be acceptableat a later time, due to advances in knowledge and technology. For instance, using quartz crystal clocks, and more recently atomic clocks, theU.S. NationalBureau ofstandards routinely measures the fluctuations in the spin angular frequency of the earth, which used to be assumed a constant. Such a measurement amounts to assuming that the observed fluctuations are due to the earth and not the quartz clocks. Any ‘corrections’ included in the definition of the standard must be considered as part of the definition, and not as justified by a theory. The relative instability of two clocks is a specific quantitative parameter that is of direct significance to the measurement of time intervals by different clocks. Consequently there is no arbitrariness in deciding which clocks have the least relative instability; in contrast to the subjective, far more difficult judgement that must be made in deciding which solitary type clock leads to the most natural and simplest physical laws. Moreover the use of several clocks in setting up the standard, reduces considerably the peculiarities of any particular clock. The above commentsplus the fact that comparison type atomicstandards are several orders of magnitude more stable than the astronomical standard, leave
VI
1,2]
CLOCKS
51
no doubt as to which type of standard is preferable, both in principle and practice. In this chapter we shall give a thorough treatment of both the definition and use of the comparison type standard (Basri [1965] Sec. 2). The subject of time standards is quite complex, and is in an active state at present, because of recent developments concerning atomic standards and related systems. The definition of the astronomical standard involves detailed knowledge of classical mechanics, astronomy, and properties of the earth, whereas atomic standards involve quantum mechanics, modern physics, and electronics. An excellent summary of the situation with regard to the astronomical standard is given by Munk and MacDonald [1960], and a review of technological developments pertinent to atomic time standards up to September 1963 is given by Mockler [1964]. We shall not get involved here in the practical details of defining a time standard, but will concentrate only on the essential ideas of what constitutes a standard. A basic knowledge of mathematical statistics, such as that presented by Cramer [1946] and Brownlee [1960], is helpful in understanding Secs. 3 and 6. 2. Clocks
Regardless of what is the actual mechanism of a clock, the common feature of all clocks is that they have a minute hand that generates events by coinciding successively with the marks of a dial, and counts the events it generates by its angular position and the position of the hour hand around the dial. The events can be thought of as produced at one particle, namely the tip of the minute hand. For our purposes, it is even better to think of the tip as stationary and the dial as rotating. In the case of a quartz or atomic clock, the tip of the minute hand is replaced by the end of the output lead, which delivers an electromagnetic signal. The signal can be used to operate an electronic counter that both exhibits and counts the events generated by the clock. Thus any clock A can be thought of as a physical system that generates and counts a sequence of events al, ...,u, at a particle P, called the output particle. To be more specific, we take P to be part of A ( P c , A ) and assume that a,, . ..,anare elements of the world line of P(a,, . . . , u , E W ( P ) ~that ) are ordered by the relation < H ( a , < H . . .
52
[VI
CLOCKS AND TIME INTERVALS
2,3
of A that occur between a, and a, on the world line of P. For a watch A , the
particles X are the marks around the dial that disappear and reappear as the tip P of the minute hand sweeps by them.
D1.
( a l , ..., a , 929P , A)H for a 1, .. ., anE w ( P ) H A a1 < H . ..
*
x,
(andHXn
XngHa~)
€ ~ ( P ) AH(3 x)x#P A X C H AA ( U d ~VxX ~ H U ) A a , < , u < H a , - + u E { a , , ..., a,,}. A (vU).U
‘(a,, ...,a,, .Y+ P , A)H’reads ‘al, ...,a, is a sequence of events at P associated with A’. Next we define a clock (not yet standard) to be an object A having an output particle P ( P c H A )such that given any event u on P , and any positive integer k , there exists a sequence of events xl, ..., xk,, at P associated with A , such that u occurs between x 1 and x ~ ( x , < ~ u < ~ This x ~ ) insures . that a clock can generate at P a sequence of events starting at any time and of any length. Moreover, for any two events u, w on P such that u precedes W ( U < ~ ~ W ) , there exists a positive integer n and events y l , ...,y,+ such that
(Yl,...,Yn+i
P , A ) H ,Y ~ < H V < H Y ~and , Y,,
i.e., there exists a sequence of events on P that brackets u and w. The output particle is unique, in the sense that if X , Y are two particles on which occur sequences of events associated with A , then the world lines of X , Yare the same ( X = Y v X;;, Y v Y;;,X). 02.
(AVGP), for P c , A A ( V U , ~ ) . U E W ( P ) , Ak E 9 . . - , x k +,)(x1, -..)x k + l s p P , A)H A X 1 < H u < H +(I A ( v V , W).V,
(Yl, A
...,4’n+l
~aplA)HAyl~HU
(v U l , . . ., urn,211, .. .)On, x,Y ) . ( U , , . ..)
A (U1,
x2:
W E ~ ( P ) H A U < H W ~ ( ~ ~ ) ~ E ..-,J’,+l) ~ A ( ( ~ ~ ~ ,
z g x,A)H
..., U, .%g Y , A),-+X= Y V xv;;*Y V Y T X .
‘(AVL‘P),’ reads ‘ A is a clock with output particle P ’ . 9is the class of positive integers. 3. Relative instability To define a standard clock, we need to know first the meaning of ‘comparison of two clocks’. We say that clocks A , B are compared, starting with event a
VI
31
53
RELATIVE INSTABILITY
of A and b of B, and to m events of A there correspond n events of B, [(A,a,mVB,b,n)H] if and only if the output particles A', Y of A , B coincide into a single particle Z ( X , Y @ . H Z ) ,and there exist events ul, ...,
u l , ...,u,+ such that a,u,, ...,u,,,+ and h,vl, ...,u,+ are sequences associated with A and B, respectively, b is between a and ul(a<,lb<,u,), and u, is between u,, and u,~,+( U , , , < ~ ~ V , < ~ U , , , + By allowing b and u, to fall within two successive events of A , instead of letting b coincide with a and u, with urn,we are in effect neglecting time intervals of duration less than the interval between two successive events of A . This error can be minimized by increasing the number of events of A that correspond to a certain number of events of B.
( A , a , m V B , b, n)H f o r m ,n €9 A ( 3 X , Y , Z > ( A % / X ) , A (B%?fY)HA X , Z A (3 u 1, . . urn + 1 , ut > . . . > u,+ 1) (0, u 1, .. .)urn + 1 .%(/Z , A)H A (b, u1 ...>un+ 1 .4"r(/ z, B)H A a BHb< H u 1 A um
DI.
.$
9
+
Fig. 3.
Comparison of clocks
+
54
CLOCKS AND TIME INTERVALS
[VI
3
From the dataf l , ...,A,we can calculate the sample moments : k
1
where J' is the sample mean, and S is the sample standard deviation. If the value of a random variable X ranges over the class of real numbers, and P(P} is the probability of F, then the function P ( x ) defined by (CramCr [1946] p. 167): P{x < X < x dx} = P(x)dx, (3)
+
is called the probability density ,function (PDF) of the random variable X. Since the probability that X has some value is 1, we must have the normalization condition : fm
J
-m
P(x)dx = 1
(4)
Let f ( x ) be a function integrable over (- co,co). The integral (CramCr [1946] p. 170): m
(f(X)>=
J f ( x >p >.(
dx
-x
(5)
9
is called the true (population) mean or expectation value of the random variablef(X).t With the help of (5), the (true) moments off, are defined by: P- = (fi>>
( ( A - I*>">
> 1)
(6) r 7 = JP-2, (7) where p is called the mean and r7 the standard deviation. We are assuming here& has the same PDF for all values of i, i.e., fl, ...,f, are identically distributed. If the PDF of& is known, it is possible to calculate confidence limits for p and 6.For example, if the distribution is normal (CramCr [1946] p. 208), i.e., ~m
=
9
(x
P (x) = (271)-' exp (- +x'>,
(8)
t For a discrete variable, such ash, Eq. (3) becomes ' P { X = x , } = P (n)', Eq. (4) becomes z,%oP(n) =1', and Eq. ( 5 ) becomes: < f ( X ) >=Z',"=O f ( x n ) P (n).
VI
31
55
RELATIVE INSTABILITY
then (Cramtr [1946] pp. 382, 387, 518; Brownlee [1960] pp. 219, 229):
P{f- t + , S / J C l < p
(9) (10) where t, and xf are the p percent values of Student’s and x z distributions of k - 1 degrees of freedom (Cramtr [1946] pp. 233-241). The expressions inside the curly brackets are of the form ‘ X < C< Y ’ , where X,Y are random variables, called the confidence limits, and C is a constant. The sentence ‘P{X< C< Y }=1-p’ states that the probability that the confidence interval ( X , Y ) covers the number C is 1-p, i.e., in a large number of trials, the fraction of times that C falls within ( X , Y ) is 1- p ; it does not state that the probability that C falls within (X, Y ) is 1-p, because this probability is either one or zero, since C either does or does not lie within ( X , Y). If the PDF offi is not known, we can still calculate confidence limits for p and c provided the number k is large enough for the central limit theorem (Cramtr [1946] p. 213) to be applicable. According to this theorem, if p and c exist, then regardless of the PDF, lim P { p - g p o / J k
k+ m
+ g p a / J k } = 1- p ,
(11)
where g p is the p percent value of a normal (Gaussian) deviate (Cramtr [1946] p. 211), i.e.,
P{lfi - pI < g,a}
= (2n)-*
‘f’exp (- +x2) dx
=
1- p .
- QP
(12)
Moreover, if p4 also exists, then (Cramtr [1946] p. 348)
lim P {c’ - g p [(p4 - c4)/k]* < S’ < cz + g p [(p4 - c4)/k]+}
k+ m
=
1- p .
(13) In effect, (11) and (13) state that for large enough k , f and S2 behave as though they were normally distributed. Confidence intervals can be derived from (11) and (13) (Cramtr [1946] pp. 366, 511), and the result is that for sufficiently large k ( k z 1 0 is usually large enough (Cramtr [1946] p. 202)), P{f- g,S/Jk-1 < p
(14) (15)
Notice that S4 is needed in (15), but not in (10); a small price to pay for not knowing the PDF.
56
CLOCKS AND TIME INTERVALS
[VI
3
Going back to our experiment, sincefi=n,/rn, we expect: p = (fi)
=
N/M
(16)
to be the true frequency ratio of the clocks, and (r to be the relative instability. Thus the sentence 'The clocks A , B have the frequency ratio M j N and relative instability o [ ( A , B %?eJM / N , o)~]'means: Clocks A , B can be compared at any time, and for as long as desired, Le., for any event u of A and any integer m, there exists an event u of B and integer n such that (A,u,mVB,u,n),. Moreover, if fi, ...,Aare the results of the experiment described after D 1, and k is large enough for the central limit theorem to hold, then [Eqs. (14), (15)] the probabilities that the confidence limits fk g,S/,/(k- 1) include the mean N / M , and S 2 g,[(S, - S4)/(k- 1)]+include the variance 02, are both equal to 1- p . More precisely, for any positive real number E , there exist three integers q,r,s such that for any integers k (number of trials), m (length of comparison), and j (length of gap), m 2 s, j2 q (to make ni independent) ; and k > r ( t o m ak e thecentrallimit theoremapplicable)iinply: ifu,, ..., ui(,,,+ j ) - j + is a sequence of events of A ; ,'A Y are the output particles of A , B ; and f l , ...&A= ni/m) are the results of the comparisons (A,ul,mV~B,u,,n,),, ( A , u ( t n + j ) + I , m y & ~ z , ~ z ) H. . ,. ) ( A ,u(k- 1 ) ( n l + j : + 13m'q.B, l'k,nk)ii; the11 for any real number p between 0 and 1, the probability that the confidence limits fk g,S/,/(k - 1) include M / N , and the probability that S k g,[(S4 - S4)/ ( k - 1)]* include 02,are both equal to 1 - p within an error c. The following definition is just a formal expression of the above idea5 (9is the class of real numbers):
VI
41
STANDARD CLOCKS
57
4. Standard clocks
Even if the relative instability of two clocks is small enough to suit our purposes, there is still the possibility that their rates may change together in the same way, so that we could not detect this change by comparing them with each other. Two pendulum clocks having the same thermal coefficient of expansion, and identical twin quartz crystal clocks cut from the same blank and aging the same way can exhibit this behavior. Fortunately, the delinquincy of such pairs can always be detected by comparing them with clocks of different upbringing. Once it is detected, something can always be done to cope with it, such as by controlling the environment or incorporate servomechanisms that keep them on the right path. This can be done without inhibition by fear of circularity for using physical theory to make corrections of instruments that themselves determine the form of the theory. The reason is that ultimately the clocks are treated as black boxes whose merits are judged by how they behave under comparison, and not by how they are constructed. Although two clocks may be constructed independently from different materials, there is still a chance that they may stray off together due to statistical fluctuations. Both this effect and the previous one due to similar construction, can be practically eliminated by increasing the number of clocks that must agree with each other. We therefore define a standard clock as an element of a set of at least rT clocks, any two of which have frequency ratio 1 and relative instability less than a certain amount C J ~ The . numbers rT,oTare agreed upon by a standards committee; rT should be at least 3, and C J for ~ atomic clocks is of the order 10-l2. D1.
( A Y%?P), f o r ( A V L P ) , A (3 r ) A 2 < r 5 rTA (3 XI, . . ., X,) # (XI, . .. , X,) A P E { X , , . .., Xr> A ( ~ x Y, ) . x # Y A X , Y E { X ,,..., X r } +(3 C J ) ( XY, V7(d l j l , c ) H A CJ I CJT.
‘(AY’gP),’ reads ‘ A is a standard clork (SC) having output particle P ’ . The number r r is called the reliability index, because the larger rTis, the more reliable we feel our standard is. The sentence ‘ # ( X , , ..., X,)’ means ‘all the X’s are different from each other’.
02.
Y%(A),{ f o r
03.
a,,
..., a,,.Y%,A
(3 x ) ( A L Y v X ) H . for
(~X)(A.Y%X)HAU ...,~u, , E W ( X ) H .
58
CLOCKS AND TIME INTERVALS
[VI
4,5
Two SC’s may not belong to the same family, and thus their frequency ratio may be different than unity. The following definition helps us express the idea that the SC’s A , , ...,A , belong to the same family. 04.
&.Y%?(A,,..., A,)H for y g ( A l ) H A ... A ~ y g ( A n ) H A # ( A , , ..., A , ) A ( V X , Y ) . X # Y A X , Y E{ A l , ...,A , } -+(30)(x, Y%‘~>I/I,o)~Ao<~~.
‘€Y%?(A,,...,A,)w’ reads ‘A1, ...,A , are equivalent standard clocks’. There are four different interactions known in physics: strong, electromagnetic, weak and gravitational. Typical examples of these interactions are, respectively, the nuclear force, the force between charged particles, the force responsible for P-decay of a nucleus, and the force between two masses. The order ofmagnitudeoftheseforcesisroughlyin theratios 1 :10-’4:10-40. A question which has been of interest in the last few years is: suppose we have two families of SC’s, each operating on the basis of a different kind of interaction, will a member of one family agree with a member of the other family? Dicke ([I9641 p. 14) and Finzi [1961] give an affirmative answer for strong and electromagnetic interactions, but leave the question open for weak and gravitational interactions.
PI.
c?,u%?(A, B)HA 8 u % ? ( B ,C ) H - + b . y % ( Ac)fi. ,
5. Time metric
A class %? is called a metric space if to every pair of elements of %? is assigned a real number z(a, b) having the following properties :
(9
(ii) (iii) (i.1
T(a, b)>0, z ( a , b)=O++a=b, r ( a , b)=T(b,a), T ( a , c) < z(a, b ) T (b,c).
+
The function t is called a metric in the space %?. With the help of SC’s we introduce a measure on 8, that has the same basic properties as a metric. A time interval between two events a, b is measured by an S C , A , by letting a, b occur on the output particle of A , and counting the events generated by A between a and b.
D1.
s(A;a,b;n), for n d A a , b . S P q , A A (3 X , ~ 1 . ..* , un+ 2) (U I ...)U n + 2 %’/ X , A),, 9
VI
51
59
TIME METRIC A :u1
<
v -u1 d < H u 2
hUn+1
A
un+ 1 < H a
2.
't(A;a,b;n),' reads 'The time interval between a,b by clock A is n'. In order to define the value of the time interval, we need to prove first that this value is unique. In T1 we prove there exists at least one value, and in T2 that there exists at most one value.
T1.
a, bYqHA+(3 n)'C(A;a , b ; n ) H .
Proof.
04.(3,l):Ant +(3 X)(A%/ X ) , 02.2+(3 n)n €Y A ( 3 U 1 , ..., U n + 2 ) ( U 1 , ..., U,+2 %a X , A ) H A : 1
T2.
(3 n) 'C ( A;a , b ;l l ) ~ . 01,02.(1.2),TV2.9,TIV(4.5,4.3): 7 ( A ;a, b ;m)H h 'C ( A ;a, b ;n ) H +(j X,u1,* * . , u m + 2 ,~ 1 ...,vn+2)* , x,A ) H A (u1, ...)un+ 2 ( u ] ,. ..) urn+ 2
Proof.
A
(u 1 < H a
*H...
A 01 < H a < H V 2
*
x,A ) H 2
1
2'
v .u, < H ~ < H ~ ~ < , . . . < H ~ ~ + ~ < H ~ < H U , + Z A 01 < H b
*
1
2)
TV2.7,TIV4.3+(3X,u1, ...,u , + ~ , v ] ..., , u,+~). (u1, . . . , u m + 2
~ ~ X , A ) H A ( U ...,un+2 l ,
A(u1
X,A)H
vv1
+ 1
A
T3.
a , b 9 F H A + ( 3 ! n ) z(A; a , b ; n)H.
Proof.
T1,2;D10.
02.
z,(a, b ) H
for
(7
n ) 7 ( A;a, b ;n ) ~ .
zA(a,b)His the value of the time interval between a and b by clock A .
T4.
a , b%%HA+.'C,(U,
Proof.
T3,T4.3,02.
b)~=n++'C(A;a,b; n)~.
60
CLOCKS AND TlME INTERVALS
[VI
5
We now proceed to prove properties of the measure sA(a,b)If analogous to the metric properties (i)-(iv).
T5. Proof.
a , bY%?HA +t~(a, b), 2 0.
Tl:Ant+(3n)z(A;u,b;n), T4, D1 +(I n ) sA( a , b)If= n A II 2 0 T5.6 +Con.
Thus property (i) is satisfied. In T6 we prove that if a and b coincide then zA(a,b), =O.
T6. Proof.
a , brY’VI,Ar\a=,b-,TA(a, b),,=O. T1,4:Ant+(I n ) z A ( U , b),=n AUXHb D1 -+(II I > T , ( ~ , b)H= n A n 1 = 1 T5.6 +Con.
+
In T7 we prove that if z,(a, b)H= 0, then a, b occur between two successive events of clock A . We cannot prove that n and b are coincident, unless the events of A are so close together that it is no longer possible to resolve a and 6 .
T7.
a, ~.Y%?~*AA T A ( u , b)l,=O
Proof.
+ ( ~ X , ~ , U ) ( ~ , U , ~ P ~ ~ / X , b
The last two theorems take the place of property (ii). If the spacing between the events of clock A is taken small enough, then a,b in T7 may be considered coincident, and property (ii) is satisfied with ‘ = ’ replaced by ‘.-II / ’.
The following theorem shows that the symmetry property (iii) is satisfied :
T8.
a, b.4PFHA- + z A ( ab)H , = z,(b,
Proof.
D 1 :T ( A;a , b ; n),++z(A:b, a ;I I ) ~ . (l), T1.14, T4: Ant+(V r I ) . ? ~ ( ab)tI , = II++TA(b, T5.4 +Con.
=
(1)
In the next three theorems, property (iv) is also proved, and thus z,(a,b), is a metric aside from the slight deviation from (ii).
T9.
a , b, c Y V H A A (a
Proof.
T l , D l : A n t + ( 3 X , u , , ..., U , , , + ~ , U ~..., , u,+~) (U1, . . ., u,,+ 2 %r/x,A)t, A (01 .. ., U,+,,sr/x, A)H
-fz,4(a, C)H=TA(a,
b ) H +T.4(b,
c<,,b<,,u) ‘)If‘
I
VI
5,6]
COMPARISON OF TIME INTERVALS
61
6. Comparison of time intervals
Due to statistical fluctuation of the period of a clock, the time metric zA(a,b)Hhas a statistical error associated with it, which we calculate in this section. Once this error is estimated, it is possible to compare the values of different time intervals. As a first step we define the mean of a time interval as the arithmetical mean of several values of the same interval measured simultaneously by different SC's with coincident output particles. More precisely, the mean ( ~ , ( a , b ) ~is)defined as the real number t satisfying the following condition : Given a positive E and 6 between 0 and 1, there exists an integer r such that for all integers k larger than r , and any equivalent standard clocks X , , ..., X,,
62
CLOCKS AND TIME INTERVALS
[VI
6
if t,,(a, b)H= n,, . .., zxk(a,b)H= nk, then the probability that the difference between the mean l z = kl ti n i ’ 1
and the number t, is larger than E , is 16, i.e., ti converges in probability to t (CramCr [1946] p. 252). According to Khintchine’s theorem (the weak law of large numbers) (Cram& [1946] pp. 253-254), this happens if n, are independent identically distributed random variables with a finite mean.
f o r ( 7 ~ ) ( V ES).F, , 6 E LZ A & >o A O S 6 < 1+(3 r ) r €9 A (V k). k d A k2r +x1, ...)x,,n,, ...)n k ) . B Y V ( X , , ... X& A zx,( a , b)Ii= n I A . . . A z x k( a , b)H= nk-+ P{ I? -i t 1 > E } I 6. (z,(a, b),) is called the mean of z,(a, b)H.
D1.
(T,(u, b),)
)
02.
V{t,4(a, b)H}
for
b)H-
b)H))z).
V{z,(a,b),} is known as the variance of t,(a,b),,. We now postulate the existence of the mean and the variance of a time interval.
P1.
(3 n)z,(a, b ) H = f l -+ (3! t ) (T, (a, b ) ~=)t A v{Z, ( Q , ~ ) 1 H 5 tit.
K is related to oT (defined in 04.1) by a constant obtained as follows: ~T=~(fi)=~(n~/m)=~(ni)/m=((ni>2/~)[~(ni)/(~i>~] =(N/Mm)+2ic. The factor 2 multiplying ic stems from the fact that ~ ( n ,characterizes ) two clocks, whereas ic2 is the variance per unit time of a single clock. Rarely do we measure a time interval by more than one clock. We can set a lower bound on the probability that this single value deviates from the mean by a certain amount, with the help of P1 and Tchebycheff’s inequality (Cramer [1946] p. 182).
Ti. Proof.
ZA ( a , b)H = f? A (t,
( a , b), ) = t
P1,Tchebycheff’s inequality.
--f
P { In - t I < ic (tip)’ } 2 1- p .
This states that the probability that In-il is less than ic(f/p)+,is larger than 1-1.’. T1 is valid regardless what is the PDF of n. If the PDF is known to be normal, then ‘p-’’ inside the square bracket in TI is replaced by the p per-
VI
61
63
COMPARISON OF TIME INTERVALS
cent value of a normal deviate ‘g,,’.
For instance, for p =
p - * = 10, but
9,%3.
In the same way, we can make a probability statement about the difference between two time intervals.
T2.
tA(a, b)H= I I
A
z E(c, d), = n2 A 8 y g (A , B)H
A (T,(a,b)H)=ti A ( T ~ ( c , d ) ~ ) = t 2 +P{ K . 1 - n,)-(t, - t 2 ) l < X[(tl + t2)/Pl’>2 1 -P.
Proof. P1, Tchebycheff’s inequality, and the fact that the variance of the difference of two independent random variables, is the sum of the variances. Since (tl + t 2 ) is not known, it is desirable to estimate it by (nl + n 2 ) . This is possible, since K<< 1 and, as above,
p (l(111 T3.
TA(U,
+ 772) - ( t l + t2)l < K [ ( t l + t2)/p]+}2 1 - p . b ) ~ = n ,A T s ( C , d ) ~ = n zA b . y g ( A , B)H
A (TA ( a , 6 ) H ) = t 1 A ( T g ( C ,
Proof.
’P{ I(n1- n,)-(t, T1,2.
d)H) = t 2 A UT << 1 + n2>/Plt) = 1 -P.
- t2)l <.[(.I
With the help of T 3 , we can now give a reasonable definition of theequality of two time intervaIs ~ ( ab), and z(c, d ) . If a=,b and c x H d ,both intervals are zero by T5.6, and are therefore equal. They are also equal if a x H c and h=Hd, or a x H dand b x H c .Otherwise, we can only state that if the difference between them is less than the error indicated in T 3 , the probability they are equal ( t l = t 2 ) is about 1- p , i.e., they are equal at a loop% confidence level. 03.
[a, b=c, d ] , , ~f o r a x H bA c x H d .V . a=cIIc A b x H d . V . UXHd A bXHC. v ( 3 X , Y , m , n ) G “ Y W ( X , Y ) HA T ~ ( u b, ) H = m A z X ( c ,d),= A I m - n I < K [( m + n ) / p ] + .
‘ [ a ,b = c, d],,,Ff’reads ‘(a,b) is p-equal to (c, d)‘.
n
We can also give a similar definition for one interval being less than another.
64
CLOCKS AND TIME INTERVALS
[vr 6
‘[a,b < c, d],, I,’ reads ‘(a,b) is p-less than (c, d)’. We now prove few basic properties of 0 3 and 0 4 . T4. Prvof.
a, b E &,+[a, a = 6 , b]p,t, 0 3 , PIV3.4, T2.15.
T5. Proof.
[ a , b=c, d ] , , , , t t [ b , n = c , d ] , , , ~ [ a , b=d,c],.,iC-’[~,d=a, b l p . ~ . 03.
A
[ a , b = b, alp,,,.
Notice that althoughp-equal is reflexive and symmetric, it is not transitive, i.e., [a,b=c,d],,, and [c,d=e,f],,,, d o not imply [ a , b = e , f ] , ., If there were n o errors, the transitive law would have been valid, but unfortunately things in practice are not as tidy as in the ivory tower. However, the transitive law does hold for y-less. T6. Pmof.
[ a , b
A
[Cd
b
The following theorem leads to the usual result that two time intervals m,n are related in one of three different ways: mn. T7. Pro0f.
(3 X , Y,Iw,~)&YV(X, Y),r\t,(~,b),=m~z,(~,d),=n + [ ~ , b = c , d ] , , , v [ a , b
CLOCKS A N D TIME INTERVALS
1. Introduction
The last two chapters are concerned with the topological structure of the class of events (without regard to the length of time intervals). In this chapter, clocks are used to define a time metric, which forms the basis of time geometry. A clock is a physical system that generates and counts a sequence of events at a particle, called the output particle; the output particle of a watch is the tip of its minute hand, and of an atomic clock is the end of its output lead. To measure the time interval between two events a , b by a clock, a, b must occur on the world line of the output particle of the clock, and the number of events generated by the clock between a and b is the value of the interval between them. If no limitation is imposed on the type of clock used, the description of physical phenomena obtained in this way, is at the mercy of the capricious behaviour of the clock: if the period of a pendulum measured by such a clock is found to vary, is the variation due to the pendulum or to the clock itself? Consequently, if a reasonable description of physical phenomena is desired, it is necessary to impose some limitations on the clocks used. In the general theory of relativity, there is a great deal of freedom in the choice of a (coordinate) clock. However, the rates of all clocks are specified relative to coincident standard clocks (by the metric coefficient g,,), and without standard clocks it is not even possible to measure the space-time line element ds. How are we to select a clock? There seem to be two different procedures, both of which are used in practice: (1) Try different clocks, and find out the kind of description of physical phenomena they lead to. The clock that gives the description which is considered by respectable physicists to be the simplest and most natural, is the clock that is crowned to be a standard clock. All other clocks are judged by comparison with it. Until the revolution of the atomic clocks, the king has been the astronomical clock. It has been a 49
50
CLOCKS AND TIME INTERVALS
[Vl 1
benevolent ruler and rendered much service to physics, but alas times do change, and the collective leadership of atomic clocks is taking over. There are very good reasons for this peaceful revolution, which will become apparent before this section is over. (2) Intercompare many clocks, and select the ones that exhibit the least relative instability. Comparison of two clocks means the coincidence of the output particles of the two clocks, and the measurement of the frequency ratio of the number of events generated by one clock to the number of events generated by the other clock between any two end events. The clocks that have the most stable ratio, i.e., the ratio with the smallest amount of fluctuation, are considered to be the clocks with the highest relative stability. This is precisely the procedure used in the selection of atomic time standards. Clearly, the more clocks that agree with each other, the better we feel about their selection. Which type of time standard is better, the solitary type selected by the first procedure or the comparison type selected by the second procedure? There is no getting around the fact that the solitary type standard is arbitrary. It cannot be justified by any theory, because such a theory must involve the concept of a time measure, which depends upon the existence of a standard, which we are trying to justify. The only justification for a solitary type standard is that the description of nature it leads to is acceptable. Unfortunately, what is acceptable at one time may not be acceptableat a later time, due to advances in knowledge and technology. For instance, using quartz crystal clocks, and more recently atomic clocks, theU.S. NationalBureau ofstandards routinely measures the fluctuations in the spin angular frequency of the earth, which used to be assumed a constant. Such a measurement amounts to assuming that the observed fluctuations are due to the earth and not the quartz clocks. Any ‘corrections’ included in the definition of the standard must be considered as part of the definition, and not as justified by a theory. The relative instability of two clocks is a specific quantitative parameter that is of direct significance to the measurement of time intervals by different clocks. Consequently there is no arbitrariness in deciding which clocks have the least relative instability; in contrast to the subjective, far more difficult judgement that must be made in deciding which solitary type clock leads to the most natural and simplest physical laws. Moreover the use of several clocks in setting up the standard, reduces considerably the peculiarities of any particular clock. The above commentsplus the fact that comparison type atomicstandards are several orders of magnitude more stable than the astronomical standard, leave
VI
1,2]
CLOCKS
51
no doubt as to which type of standard is preferable, both in principle and practice. In this chapter we shall give a thorough treatment of both the definition and use of the comparison type standard (Basri [1965] Sec. 2). The subject of time standards is quite complex, and is in an active state at present, because of recent developments concerning atomic standards and related systems. The definition of the astronomical standard involves detailed knowledge of classical mechanics, astronomy, and properties of the earth, whereas atomic standards involve quantum mechanics, modern physics, and electronics. An excellent summary of the situation with regard to the astronomical standard is given by Munk and MacDonald [1960], and a review of technological developments pertinent to atomic time standards up to September 1963 is given by Mockler [1964]. We shall not get involved here in the practical details of defining a time standard, but will concentrate only on the essential ideas of what constitutes a standard. A basic knowledge of mathematical statistics, such as that presented by Cramer [1946] and Brownlee [1960], is helpful in understanding Secs. 3 and 6. 2. Clocks
Regardless of what is the actual mechanism of a clock, the common feature of all clocks is that they have a minute hand that generates events by coinciding successively with the marks of a dial, and counts the events it generates by its angular position and the position of the hour hand around the dial. The events can be thought of as produced at one particle, namely the tip of the minute hand. For our purposes, it is even better to think of the tip as stationary and the dial as rotating. In the case of a quartz or atomic clock, the tip of the minute hand is replaced by the end of the output lead, which delivers an electromagnetic signal. The signal can be used to operate an electronic counter that both exhibits and counts the events generated by the clock. Thus any clock A can be thought of as a physical system that generates and counts a sequence of events al, ...,u, at a particle P, called the output particle. To be more specific, we take P to be part of A ( P c , A ) and assume that a,, . ..,anare elements of the world line of P(a,, . . . , u , E W ( P ) ~that ) are ordered by the relation < H ( a , < H . . .
52
[VI
CLOCKS AND TIME INTERVALS
2,3
of A that occur between a, and a, on the world line of P. For a watch A , the
particles X are the marks around the dial that disappear and reappear as the tip P of the minute hand sweeps by them.
D1.
( a l , ..., a , 929P , A)H for a 1, .. ., anE w ( P ) H A a1 < H . ..
*
x,
(andHXn
XngHa~)
€ ~ ( P ) AH(3 x)x#P A X C H AA ( U d ~VxX ~ H U ) A a , < , u < H a , - + u E { a , , ..., a,,}. A (vU).U
‘(a,, ...,a,, .Y+ P , A)H’reads ‘al, ...,a, is a sequence of events at P associated with A’. Next we define a clock (not yet standard) to be an object A having an output particle P ( P c H A )such that given any event u on P , and any positive integer k , there exists a sequence of events xl, ..., xk,, at P associated with A , such that u occurs between x 1 and x ~ ( x , < ~ u < ~ This x ~ ) insures . that a clock can generate at P a sequence of events starting at any time and of any length. Moreover, for any two events u, w on P such that u precedes W ( U < ~ ~ W ) , there exists a positive integer n and events y l , ...,y,+ such that
(Yl,...,Yn+i
P , A ) H ,Y ~ < H V < H Y ~and , Y,,
i.e., there exists a sequence of events on P that brackets u and w. The output particle is unique, in the sense that if X , Y are two particles on which occur sequences of events associated with A , then the world lines of X , Yare the same ( X = Y v X;;, Y v Y;;,X). 02.
(AVGP), for P c , A A ( V U , ~ ) . U E W ( P ) , Ak E 9 . . - , x k +,)(x1, -..)x k + l s p P , A)H A X 1 < H u < H +(I A ( v V , W).V,
(Yl, A
...,4’n+l
~aplA)HAyl~HU
(v U l , . . ., urn,211, .. .)On, x,Y ) . ( U , , . ..)
A (U1,
x2:
W E ~ ( P ) H A U < H W ~ ( ~ ~ ) ~ E ..-,J’,+l) ~ A ( ( ~ ~ ~ ,
z g x,A)H
..., U, .%g Y , A),-+X= Y V xv;;*Y V Y T X .
‘(AVL‘P),’ reads ‘ A is a clock with output particle P ’ . 9is the class of positive integers. 3. Relative instability To define a standard clock, we need to know first the meaning of ‘comparison of two clocks’. We say that clocks A , B are compared, starting with event a
VI
31
53
RELATIVE INSTABILITY
of A and b of B, and to m events of A there correspond n events of B, [(A,a,mVB,b,n)H] if and only if the output particles A', Y of A , B coincide into a single particle Z ( X , Y @ . H Z ) ,and there exist events ul, ...,
u l , ...,u,+ such that a,u,, ...,u,,,+ and h,vl, ...,u,+ are sequences associated with A and B, respectively, b is between a and ul(a<,lb<,u,), and u, is between u,, and u,~,+( U , , , < ~ ~ V , < ~ U , , , + By allowing b and u, to fall within two successive events of A , instead of letting b coincide with a and u, with urn,we are in effect neglecting time intervals of duration less than the interval between two successive events of A . This error can be minimized by increasing the number of events of A that correspond to a certain number of events of B.
( A , a , m V B , b, n)H f o r m ,n €9 A ( 3 X , Y , Z > ( A % / X ) , A (B%?fY)HA X , Z A (3 u 1, . . urn + 1 , ut > . . . > u,+ 1) (0, u 1, .. .)urn + 1 .%(/Z , A)H A (b, u1 ...>un+ 1 .4"r(/ z, B)H A a BHb< H u 1 A um
DI.
.$
9
+
Fig. 3.
Comparison of clocks
+
54
CLOCKS AND TIME INTERVALS
[VI
3
From the dataf l , ...,A,we can calculate the sample moments : k
1
where J' is the sample mean, and S is the sample standard deviation. If the value of a random variable X ranges over the class of real numbers, and P(P} is the probability of F, then the function P ( x ) defined by (CramCr [1946] p. 167): P{x < X < x dx} = P(x)dx, (3)
+
is called the probability density ,function (PDF) of the random variable X. Since the probability that X has some value is 1, we must have the normalization condition : fm
J
-m
P(x)dx = 1
(4)
Let f ( x ) be a function integrable over (- co,co). The integral (CramCr [1946] p. 170): m
(f(X)>=
J f ( x >p >.(
dx
-x
(5)
9
is called the true (population) mean or expectation value of the random variablef(X).t With the help of (5), the (true) moments off, are defined by: P- = (fi>>
( ( A - I*>">
> 1)
(6) r 7 = JP-2, (7) where p is called the mean and r7 the standard deviation. We are assuming here& has the same PDF for all values of i, i.e., fl, ...,f, are identically distributed. If the PDF of& is known, it is possible to calculate confidence limits for p and 6.For example, if the distribution is normal (CramCr [1946] p. 208), i.e., ~m
=
9
(x
P (x) = (271)-' exp (- +x'>,
(8)
t For a discrete variable, such ash, Eq. (3) becomes ' P { X = x , } = P (n)', Eq. (4) becomes z,%oP(n) =1', and Eq. ( 5 ) becomes: < f ( X ) >=Z',"=O f ( x n ) P (n).
VI
31
55
RELATIVE INSTABILITY
then (Cramtr [1946] pp. 382, 387, 518; Brownlee [1960] pp. 219, 229):
P{f- t + , S / J C l < p
(9) (10) where t, and xf are the p percent values of Student’s and x z distributions of k - 1 degrees of freedom (Cramtr [1946] pp. 233-241). The expressions inside the curly brackets are of the form ‘ X < C< Y ’ , where X,Y are random variables, called the confidence limits, and C is a constant. The sentence ‘P{X< C< Y }=1-p’ states that the probability that the confidence interval ( X , Y ) covers the number C is 1-p, i.e., in a large number of trials, the fraction of times that C falls within ( X , Y ) is 1- p ; it does not state that the probability that C falls within (X, Y ) is 1-p, because this probability is either one or zero, since C either does or does not lie within ( X , Y). If the PDF offi is not known, we can still calculate confidence limits for p and c provided the number k is large enough for the central limit theorem (Cramtr [1946] p. 213) to be applicable. According to this theorem, if p and c exist, then regardless of the PDF, lim P { p - g p o / J k
k+ m
+ g p a / J k } = 1- p ,
(11)
where g p is the p percent value of a normal (Gaussian) deviate (Cramtr [1946] p. 211), i.e.,
P{lfi - pI < g,a}
= (2n)-*
‘f’exp (- +x2) dx
=
1- p .
- QP
(12)
Moreover, if p4 also exists, then (Cramtr [1946] p. 348)
lim P {c’ - g p [(p4 - c4)/k]* < S’ < cz + g p [(p4 - c4)/k]+}
k+ m
=
1- p .
(13) In effect, (11) and (13) state that for large enough k , f and S2 behave as though they were normally distributed. Confidence intervals can be derived from (11) and (13) (Cramtr [1946] pp. 366, 511), and the result is that for sufficiently large k ( k z 1 0 is usually large enough (Cramtr [1946] p. 202)), P{f- g,S/Jk-1 < p
(14) (15)
Notice that S4 is needed in (15), but not in (10); a small price to pay for not knowing the PDF.
56
CLOCKS AND TIME INTERVALS
[VI
3
Going back to our experiment, sincefi=n,/rn, we expect: p = (fi)
=
N/M
(16)
to be the true frequency ratio of the clocks, and (r to be the relative instability. Thus the sentence 'The clocks A , B have the frequency ratio M j N and relative instability o [ ( A , B %?eJM / N , o)~]'means: Clocks A , B can be compared at any time, and for as long as desired, Le., for any event u of A and any integer m, there exists an event u of B and integer n such that (A,u,mVB,u,n),. Moreover, if fi, ...,Aare the results of the experiment described after D 1, and k is large enough for the central limit theorem to hold, then [Eqs. (14), (15)] the probabilities that the confidence limits fk g,S/,/(k- 1) include the mean N / M , and S 2 g,[(S, - S4)/(k- 1)]+include the variance 02, are both equal to 1- p . More precisely, for any positive real number E , there exist three integers q,r,s such that for any integers k (number of trials), m (length of comparison), and j (length of gap), m 2 s, j2 q (to make ni independent) ; and k > r ( t o m ak e thecentrallimit theoremapplicable)iinply: ifu,, ..., ui(,,,+ j ) - j + is a sequence of events of A ; ,'A Y are the output particles of A , B ; and f l , ...&A= ni/m) are the results of the comparisons (A,ul,mV~B,u,,n,),, ( A , u ( t n + j ) + I , m y & ~ z , ~ z ) H. . ,. ) ( A ,u(k- 1 ) ( n l + j : + 13m'q.B, l'k,nk)ii; the11 for any real number p between 0 and 1, the probability that the confidence limits fk g,S/,/(k - 1) include M / N , and the probability that S k g,[(S4 - S4)/ ( k - 1)]* include 02,are both equal to 1 - p within an error c. The following definition is just a formal expression of the above idea5 (9is the class of real numbers):
VI
41
STANDARD CLOCKS
57
4. Standard clocks
Even if the relative instability of two clocks is small enough to suit our purposes, there is still the possibility that their rates may change together in the same way, so that we could not detect this change by comparing them with each other. Two pendulum clocks having the same thermal coefficient of expansion, and identical twin quartz crystal clocks cut from the same blank and aging the same way can exhibit this behavior. Fortunately, the delinquincy of such pairs can always be detected by comparing them with clocks of different upbringing. Once it is detected, something can always be done to cope with it, such as by controlling the environment or incorporate servomechanisms that keep them on the right path. This can be done without inhibition by fear of circularity for using physical theory to make corrections of instruments that themselves determine the form of the theory. The reason is that ultimately the clocks are treated as black boxes whose merits are judged by how they behave under comparison, and not by how they are constructed. Although two clocks may be constructed independently from different materials, there is still a chance that they may stray off together due to statistical fluctuations. Both this effect and the previous one due to similar construction, can be practically eliminated by increasing the number of clocks that must agree with each other. We therefore define a standard clock as an element of a set of at least rT clocks, any two of which have frequency ratio 1 and relative instability less than a certain amount C J ~ The . numbers rT,oTare agreed upon by a standards committee; rT should be at least 3, and C J for ~ atomic clocks is of the order 10-l2. D1.
( A Y%?P), f o r ( A V L P ) , A (3 r ) A 2 < r 5 rTA (3 XI, . . ., X,) # (XI, . .. , X,) A P E { X , , . .., Xr> A ( ~ x Y, ) . x # Y A X , Y E { X ,,..., X r } +(3 C J ) ( XY, V7(d l j l , c ) H A CJ I CJT.
‘(AY’gP),’ reads ‘ A is a standard clork (SC) having output particle P ’ . The number r r is called the reliability index, because the larger rTis, the more reliable we feel our standard is. The sentence ‘ # ( X , , ..., X,)’ means ‘all the X’s are different from each other’.
02.
Y%(A),{ f o r
03.
a,,
..., a,,.Y%,A
(3 x ) ( A L Y v X ) H . for
(~X)(A.Y%X)HAU ...,~u, , E W ( X ) H .
58
CLOCKS AND TIME INTERVALS
[VI
4,5
Two SC’s may not belong to the same family, and thus their frequency ratio may be different than unity. The following definition helps us express the idea that the SC’s A , , ...,A , belong to the same family. 04.
&.Y%?(A,,..., A,)H for y g ( A l ) H A ... A ~ y g ( A n ) H A # ( A , , ..., A , ) A ( V X , Y ) . X # Y A X , Y E{ A l , ...,A , } -+(30)(x, Y%‘~>I/I,o)~Ao<~~.
‘€Y%?(A,,...,A,)w’ reads ‘A1, ...,A , are equivalent standard clocks’. There are four different interactions known in physics: strong, electromagnetic, weak and gravitational. Typical examples of these interactions are, respectively, the nuclear force, the force between charged particles, the force responsible for P-decay of a nucleus, and the force between two masses. The order ofmagnitudeoftheseforcesisroughlyin theratios 1 :10-’4:10-40. A question which has been of interest in the last few years is: suppose we have two families of SC’s, each operating on the basis of a different kind of interaction, will a member of one family agree with a member of the other family? Dicke ([I9641 p. 14) and Finzi [1961] give an affirmative answer for strong and electromagnetic interactions, but leave the question open for weak and gravitational interactions.
PI.
c?,u%?(A, B)HA 8 u % ? ( B ,C ) H - + b . y % ( Ac)fi. ,
5. Time metric
A class %? is called a metric space if to every pair of elements of %? is assigned a real number z(a, b) having the following properties :
(9
(ii) (iii) (i.1
T(a, b)>0, z ( a , b)=O++a=b, r ( a , b)=T(b,a), T ( a , c) < z(a, b ) T (b,c).
+
The function t is called a metric in the space %?. With the help of SC’s we introduce a measure on 8, that has the same basic properties as a metric. A time interval between two events a, b is measured by an S C , A , by letting a, b occur on the output particle of A , and counting the events generated by A between a and b.
D1.
s(A;a,b;n), for n d A a , b . S P q , A A (3 X , ~ 1 . ..* , un+ 2) (U I ...)U n + 2 %’/ X , A),, 9
VI
51
59
TIME METRIC A :u1
<
v -u1 d < H u 2
hUn+1
A
un+ 1 < H a
2.
't(A;a,b;n),' reads 'The time interval between a,b by clock A is n'. In order to define the value of the time interval, we need to prove first that this value is unique. In T1 we prove there exists at least one value, and in T2 that there exists at most one value.
T1.
a, bYqHA+(3 n)'C(A;a , b ; n ) H .
Proof.
04.(3,l):Ant +(3 X)(A%/ X ) , 02.2+(3 n)n €Y A ( 3 U 1 , ..., U n + 2 ) ( U 1 , ..., U,+2 %a X , A ) H A : 1
T2.
(3 n) 'C ( A;a , b ;l l ) ~ . 01,02.(1.2),TV2.9,TIV(4.5,4.3): 7 ( A ;a, b ;m)H h 'C ( A ;a, b ;n ) H +(j X,u1,* * . , u m + 2 ,~ 1 ...,vn+2)* , x,A ) H A (u1, ...)un+ 2 ( u ] ,. ..) urn+ 2
Proof.
A
(u 1 < H a
*H...
A 01 < H a < H V 2
*
x,A ) H 2
1
2'
v .u, < H ~ < H ~ ~ < , . . . < H ~ ~ + ~ < H ~ < H U , + Z A 01 < H b
*
1
2)
TV2.7,TIV4.3+(3X,u1, ...,u , + ~ , v ] ..., , u,+~). (u1, . . . , u m + 2
~ ~ X , A ) H A ( U ...,un+2 l ,
A(u1
X,A)H
vv1
+ 1
A
T3.
a , b 9 F H A + ( 3 ! n ) z(A; a , b ; n)H.
Proof.
T1,2;D10.
02.
z,(a, b ) H
for
(7
n ) 7 ( A;a, b ;n ) ~ .
zA(a,b)His the value of the time interval between a and b by clock A .
T4.
a , b%%HA+.'C,(U,
Proof.
T3,T4.3,02.
b)~=n++'C(A;a,b; n)~.
60
CLOCKS AND TlME INTERVALS
[VI
5
We now proceed to prove properties of the measure sA(a,b)If analogous to the metric properties (i)-(iv).
T5. Proof.
a , bY%?HA +t~(a, b), 2 0.
Tl:Ant+(3n)z(A;u,b;n), T4, D1 +(I n ) sA( a , b)If= n A II 2 0 T5.6 +Con.
Thus property (i) is satisfied. In T6 we prove that if a and b coincide then zA(a,b), =O.
T6. Proof.
a , brY’VI,Ar\a=,b-,TA(a, b),,=O. T1,4:Ant+(I n ) z A ( U , b),=n AUXHb D1 -+(II I > T , ( ~ , b)H= n A n 1 = 1 T5.6 +Con.
+
In T7 we prove that if z,(a, b)H= 0, then a, b occur between two successive events of clock A . We cannot prove that n and b are coincident, unless the events of A are so close together that it is no longer possible to resolve a and 6 .
T7.
a, ~.Y%?~*AA T A ( u , b)l,=O
Proof.
+ ( ~ X , ~ , U ) ( ~ , U , ~ P ~ ~ / X , b
The last two theorems take the place of property (ii). If the spacing between the events of clock A is taken small enough, then a,b in T7 may be considered coincident, and property (ii) is satisfied with ‘ = ’ replaced by ‘.-II / ’.
The following theorem shows that the symmetry property (iii) is satisfied :
T8.
a, b.4PFHA- + z A ( ab)H , = z,(b,
Proof.
D 1 :T ( A;a , b ; n),++z(A:b, a ;I I ) ~ . (l), T1.14, T4: Ant+(V r I ) . ? ~ ( ab)tI , = II++TA(b, T5.4 +Con.
=
(1)
In the next three theorems, property (iv) is also proved, and thus z,(a,b), is a metric aside from the slight deviation from (ii).
T9.
a , b, c Y V H A A (a
Proof.
T l , D l : A n t + ( 3 X , u , , ..., U , , , + ~ , U ~..., , u,+~) (U1, . . ., u,,+ 2 %r/x,A)t, A (01 .. ., U,+,,sr/x, A)H
-fz,4(a, C)H=TA(a,
b ) H +T.4(b,
c<,,b<,,u) ‘)If‘
I
VI
5,6]
COMPARISON OF TIME INTERVALS
61
6. Comparison of time intervals
Due to statistical fluctuation of the period of a clock, the time metric zA(a,b)Hhas a statistical error associated with it, which we calculate in this section. Once this error is estimated, it is possible to compare the values of different time intervals. As a first step we define the mean of a time interval as the arithmetical mean of several values of the same interval measured simultaneously by different SC's with coincident output particles. More precisely, the mean ( ~ , ( a , b ) ~is)defined as the real number t satisfying the following condition : Given a positive E and 6 between 0 and 1, there exists an integer r such that for all integers k larger than r , and any equivalent standard clocks X , , ..., X,,
62
CLOCKS AND TIME INTERVALS
[VI
6
if t,,(a, b)H= n,, . .., zxk(a,b)H= nk, then the probability that the difference between the mean l z = kl ti n i ’ 1
and the number t, is larger than E , is 16, i.e., ti converges in probability to t (CramCr [1946] p. 252). According to Khintchine’s theorem (the weak law of large numbers) (Cram& [1946] pp. 253-254), this happens if n, are independent identically distributed random variables with a finite mean.
f o r ( 7 ~ ) ( V ES).F, , 6 E LZ A & >o A O S 6 < 1+(3 r ) r €9 A (V k). k d A k2r +x1, ...)x,,n,, ...)n k ) . B Y V ( X , , ... X& A zx,( a , b)Ii= n I A . . . A z x k( a , b)H= nk-+ P{ I? -i t 1 > E } I 6. (z,(a, b),) is called the mean of z,(a, b)H.
D1.
(T,(u, b),)
)
02.
V{t,4(a, b)H}
for
b)H-
b)H))z).
V{z,(a,b),} is known as the variance of t,(a,b),,. We now postulate the existence of the mean and the variance of a time interval.
P1.
(3 n)z,(a, b ) H = f l -+ (3! t ) (T, (a, b ) ~=)t A v{Z, ( Q , ~ ) 1 H 5 tit.
K is related to oT (defined in 04.1) by a constant obtained as follows: ~T=~(fi)=~(n~/m)=~(ni)/m=((ni>2/~)[~(ni)/(~i>~] =(N/Mm)+2ic. The factor 2 multiplying ic stems from the fact that ~ ( n ,characterizes ) two clocks, whereas ic2 is the variance per unit time of a single clock. Rarely do we measure a time interval by more than one clock. We can set a lower bound on the probability that this single value deviates from the mean by a certain amount, with the help of P1 and Tchebycheff’s inequality (Cramer [1946] p. 182).
Ti. Proof.
ZA ( a , b)H = f? A (t,
( a , b), ) = t
P1,Tchebycheff’s inequality.
--f
P { In - t I < ic (tip)’ } 2 1- p .
This states that the probability that In-il is less than ic(f/p)+,is larger than 1-1.’. T1 is valid regardless what is the PDF of n. If the PDF is known to be normal, then ‘p-’’ inside the square bracket in TI is replaced by the p per-
VI
61
63
COMPARISON OF TIME INTERVALS
cent value of a normal deviate ‘g,,’.
For instance, for p =
p - * = 10, but
9,%3.
In the same way, we can make a probability statement about the difference between two time intervals.
T2.
tA(a, b)H= I I
A
z E(c, d), = n2 A 8 y g (A , B)H
A (T,(a,b)H)=ti A ( T ~ ( c , d ) ~ ) = t 2 +P{ K . 1 - n,)-(t, - t 2 ) l < X[(tl + t2)/Pl’>2 1 -P.
Proof. P1, Tchebycheff’s inequality, and the fact that the variance of the difference of two independent random variables, is the sum of the variances. Since (tl + t 2 ) is not known, it is desirable to estimate it by (nl + n 2 ) . This is possible, since K<< 1 and, as above,
p (l(111 T3.
TA(U,
+ 772) - ( t l + t2)l < K [ ( t l + t2)/p]+}2 1 - p . b ) ~ = n ,A T s ( C , d ) ~ = n zA b . y g ( A , B)H
A (TA ( a , 6 ) H ) = t 1 A ( T g ( C ,
Proof.
’P{ I(n1- n,)-(t, T1,2.
d)H) = t 2 A UT << 1 + n2>/Plt) = 1 -P.
- t2)l <.[(.I
With the help of T 3 , we can now give a reasonable definition of theequality of two time intervaIs ~ ( ab), and z(c, d ) . If a=,b and c x H d ,both intervals are zero by T5.6, and are therefore equal. They are also equal if a x H c and h=Hd, or a x H dand b x H c .Otherwise, we can only state that if the difference between them is less than the error indicated in T 3 , the probability they are equal ( t l = t 2 ) is about 1- p , i.e., they are equal at a loop% confidence level. 03.
[a, b=c, d ] , , ~f o r a x H bA c x H d .V . a=cIIc A b x H d . V . UXHd A bXHC. v ( 3 X , Y , m , n ) G “ Y W ( X , Y ) HA T ~ ( u b, ) H = m A z X ( c ,d),= A I m - n I < K [( m + n ) / p ] + .
‘ [ a ,b = c, d],,,Ff’reads ‘(a,b) is p-equal to (c, d)‘.
n
We can also give a similar definition for one interval being less than another.
64
CLOCKS AND TIME INTERVALS
[vr 6
‘[a,b < c, d],, I,’ reads ‘(a,b) is p-less than (c, d)’. We now prove few basic properties of 0 3 and 0 4 . T4. Prvof.
a, b E &,+[a, a = 6 , b]p,t, 0 3 , PIV3.4, T2.15.
T5. Proof.
[ a , b=c, d ] , , , , t t [ b , n = c , d ] , , , ~ [ a , b=d,c],.,iC-’[~,d=a, b l p . ~ . 03.
A
[ a , b = b, alp,,,.
Notice that althoughp-equal is reflexive and symmetric, it is not transitive, i.e., [a,b=c,d],,, and [c,d=e,f],,,, d o not imply [ a , b = e , f ] , ., If there were n o errors, the transitive law would have been valid, but unfortunately things in practice are not as tidy as in the ivory tower. However, the transitive law does hold for y-less. T6. Pmof.
[ a , b
A
[Cd
b
The following theorem leads to the usual result that two time intervals m,n are related in one of three different ways: m
(3 X , Y,Iw,~)&YV(X, Y),r\t,(~,b),=m~z,(~,d),=n + [ ~ , b = c , d ] , , , v [ a , b