The Museum of HP Calculators
The Museum of HP Calculators
This program is Copyright © 2007 by Jean-Marc Baillard and is used here by permission.
This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
1°) The Lorentz Transformation
2°) Composition of Speeds & Relative
Velocity
3°) Doppler Effect & Aberration
-See also "Einstein's Twin Paradox for the HP-41"
1°) The Lorentz Transformation
y'
y
| z'
|
| /
|
z |
/
|
/ |
/
| /
|/-------------------------- x'
( Ox ) // ( O'x' )
| /
O'
( Oy ) // ( O'y' )
|/-----------------------------------
x
( Oz ) // ( O'z' )
O
-We assume that O' has a constant velocity ß = ( ßx
, ßy , ßz ) with respect to ( Ox , Oy
, Oz ).
and that the event ( 0 , 0 , 0 , 0 ) has the same coordinates
in both reference-frames.
-The following program performs the conversion ( x , y , z , ct
) < > ( x' , y' , z' , ct' ) for a given event
Formulae: With r(x,y,z) and r'(x',y',z') , g = ( 1 - b2 ) -1/2 , ß = || ß || = ( ßx2 + ßy2 +ßz2 ) 1/2
r = r' + ß.g [ g/(g+1)
ß.r'
+ c t' ]
c t = g ( c t' + ß.r' )
where ß.r and ß.r' are the dot products
r' = r + ß.g [
g/(g+1)
ß.r - c t ]
c t' = g ( c t' - ß.r )
Data Registers: R00 = Dilatation Factor = g = ( 1 - ß2 ) -1/2 ( Registers R01 thru R03 are to be initialized before executing "LRTF" )
• R01 = ßx
R04 = x ( or x' if SF 00 )
R08 = the invariant c2 t2 - x2
- y2 - z2 = c2 t'2 - x'2
- y'2 - z'2
• R02 = ßy
R05 = y ( or y' if SF 00 )
• R03 = ßz
R06 = z ( or z' if SF 00 )
R07 = ct ( or ct' if SF 00 )
Flags: F00
CF 00 for the transformation ( r , ct ) >>>
( r' , ct' )
SF 00 for the transformation ( r' , ct' ) >>>
( r , ct )
Subroutines: /
01 LBL "LRTF"
02 STO 04
03 X^2
04 X<>Y
05 STO 05
06 X^2
07 +
08 X<>Y
09 STO 06
10 X^2
11 +
12 X<>Y
13 STO 07
14 X^2
15 X<>Y
16 -
17 STO 08
18 1
19 RCL 01
20 X^2
21 RCL 02
22 X^2
23 RCL 03
24 X^2
25 +
26 +
27 -
28 SQRT
29 1/X
30 STO 00
31 RCL 07
32 RCL 01
33 RCL 04
34 *
35 RCL 02
36 RCL 05
37 *
38 +
39 RCL 03
40 RCL 06
41 *
42 +
43 FS? 00
44 ST+ Y
45 FC? 00
46 ST- Y
47 RCL 00
48 ST* Z
49 ST* Y
50 1
51 +
52 /
53 RCL 07
54 FC? 00
55 CHS
56 +
57 RCL 00
58 *
59 RCL 03
60 X<>Y
61 *
62 RCL 02
63 LASTX
64 *
65 RCL 01
66 ST* L
67 CLX
68 RCL 06
69 ST+ Z
70 CLX
71 RCL 05
72 ST+ Y
73 CLX
74 RCL 04
75 ST+ L
76 X<> L
77 END
( 98 bytes / SIZE 009 )
| STACK(CF00) | INPUTS | OUTPUTS |
| T | c.t | c.t' |
| Z | z | z' |
| Y | y | y' |
| X | x | x' |
| STACK(SF 00) | INPUTS | OUTPUTS |
| T | c.t' | c.t |
| Z | z' | z |
| Y | y' | y |
| X | x' | x |
Example1: Velocity ß = ( 0.4 ; 0.5 ; 0.6 ) Event = ( x , y , z , c.t ) = ( 1 , 2 , 3 , 4 )
0.4 STO 01
0.5 STO 02
0.6 STO 06
CF 00
4 ENTER^
3 ENTER^
2 ENTER^
1 XEQ "LRTF" >>>> x =
-0.5324
RDN y = 0.0846
RDN z = 0.7015
RDN c.t = 1.6681
-We also have the dilatation factor g = R00 = 2.0851 and
the invariant c2 t2 - x2 -
y2 - z2 = R08 = 2
Example2: With the same ß = ( 0.4 ; 0.5 ; 0.6 ) Event = ( x' , y' , z' , c.t' ) = ( 1 , 2 , 3 , 4 )
SF 00
4 ENTER^
3 ENTER^
2 ENTER^
1 XEQ "LRTF" >>>> x =
6.1401
RDN y = 8.4251
RDN z = 10.7102
RDN c.t = 15.0130
Notes:
-If R08 > 0 it is a time-like event.
-If R08 < 0 it is a space-like event.
2°) Composition of Speeds & Relative Velocity
-We assume that the units are chosen so that c = speed of light = 1
y'
y
| z'
|
| /
|
z |
/ . P = Particle
|
/ |
/
| /
|/-------------------------- x'
| /
O'
|/-----------------------------------
x
O
- O' has a constant velocity u = ( ux , uy , uz ) with respect to ( Ox , Oy , Oz )
Problem1: The speed of a particle P is v = ( vx , vy , vz ) with respect to ( Ox' , Oy' , Oz' ). Calculate its speed w with respect to ( Ox , Oy , Oz ) ?
Answer: w = ( wx , wy , wz ) with
wx = [ vx + g.ux
( g u.v /(g+1) + 1 ] / [ g ( 1 + u.v ) ]
wy = [ vy + g.uy ( g
u.v
/(g+1) + 1 ] / [ g ( 1 + u.v ) ]
u.v = dot product g = ( 1 - ux2
- uy2 - uz2 ) -1/2
wz = [ vz + g.uz
( g u.v /(g+1) + 1 ] / [ g ( 1 + u.v ) ]
Problem2: The speed of this particle P is v = ( vx , vy , vz ) with respect to ( Ox , Oy , Oz ). Calculate its relative velocity w with respect to ( Ox' , Oy' , Oz' ) ?
Answer: w = ( wx , wy , wz ) with
wx = [ vx + g.ux
( g u.v /(g+1) - 1 ] / [ g ( 1 - u.v ) ]
wy = [ vy + g.uy ( g
u.v
/(g+1) - 1 ] / [ g ( 1 - u.v ) ]
wz = [ vz + g.uz
( g u.v /(g+1) - 1 ] / [ g ( 1 - u.v ) ]
-Since these formulae only differ in a few changes of sign, we can use
a unique program and a flag.
Data Registers: R00 = Dilatation Factor = g = ( 1 - u2 ) -1/2 ( Registers R01 thru R06 are to be initialized before executing "SPEED" )
• R01 = ux • R04 =
vx R07 = wx
• R02 = uy • R05 = vy
R08 = wy
• R03 = uz •
R06 = vz
R09 = wz
Flag: F01
CF 01 for the composition of speeds
SF 01 for the relative velocity
Subroutines: /
01 LBL "SPEED"
02 1
03 RCL 01
04 X^2
05 RCL 02
06 X^2
07 RCL 03
08 X^2
09 +
10 +
11 -
12 SQRT
13 1/X
14 STO 00
15 RCL 01
16 RCL 04
17 *
18 RCL 02
19 RCL 05
20 *
21 +
22 RCL 03
23 RCL 06
24 *
25 +
26 STO 07
27 *
28 X<>Y
29 ST* 07
30 FC? 01
31 ST+ 07
32 FS? 01
33 ST- 07
34 1
35 +
36 /
37 *
38 FS? 01
39 CHS
40 +
41 STO 08
42 RCL 03
43 *
44 RCL 06
45 FS? 01
46 CHS
47 +
48 RCL 02
49 RCL 08
50 *
51 RCL 05
52 FS? 01
53 CHS
54 +
55 RCL 01
56 RCL 08
57 *
58 RCL 04
59 FS? 01
60 CHS
61 +
62 RCL 07
63 ST/ T
64 ST/ Z
65 /
66 STO 07
67 X^2
68 X<>Y
69 STO 08
70 X^2
71 +
72 X<>Y
73 STO 09
74 X^2
75 +
76 SQRT
77 RCL 09
78 RCL 08
79 RCL 07
80 END
( 101 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| T | / | w |
| Z | / | wz |
| Y | / | wy |
| X | / | wx |
where w is the norm of the 3-vector w = ( wx , wy , wz )
-if CF 01 w = the composition of u and
v
-if SF 01 w = the relative speed ( with respect
to the 2nd reference-frame )
Example: u = ( 0.4 , 0.5 , 0.6 ) v = ( 0.27 , 0.37 , 0.47 )
0.4 STO 01
0.27 STO 04
0.5 STO 02
0.37 STO 05
0.6 STO 03
0.47 STO 06
CF 01 XEQ "SPEED" >>>> wx
= 0.434880406
( execution time = 3 seconds )
RDN wy = 0.553496668
RDN wz = 0.672112930
RDN w = 0.973249875
SF 01 XEQ "SPEED" >>>> wx
= -0.270737319
RDN wy = -0.301747643
RDN wz = -0.332757967
RDN w = 0.524478981
Notes:
-The composition of speeds is not commutative in the general
case.
-However, the norms w are identical
-There will be a DATA ERROR line 12 or 13 if u = || u
|| is equal to or greater than 1
-However, the program also works if v = || v || = 1 (
a photon )
and if v = || v || is greater than 1. In this case,
the particle P is a ( hypothetic ) tachyon!
-The formulas are much simpler if uy = uz
= vy = vz = 0
then wy = wz = 0 and for the
composition of speeds:
wx = ( ux + vx ) / ( 1 + ux
vx )
-If you only want to compute the magnitude w , the routine hereafter
is shorter than "SPEED"
-It uses the formulae:
w2 = [ ( u + v )2 - ( u
x v )2 ] / ( 1 + u.v )2
for the composition of speeds
where u.v is the dot product
w2 = [ ( u - v )2 - ( u
x v )2 ] / ( 1 - u.v )2
for the relative velocities
and u x v is the cross-product
Data Registers: R00 = w ( Registers R01 thru R06 are to be initialized before executing "MAGW" )
• R01 = ux • R04 =
vx
• R02 = uy • R05 = vy
• R03 = uz •
R06 = vz
Flag: F01
CF 01 for the composition of speeds
SF 01 for the relative speed
Subroutines: "DOT" & "CROSS" ( cf "Dot-product
& Cross-product for the HP-41" )
01 LBL "MAGW"
02 1
03 RCL 06
04 RCL 05
05 RCL 04
06 XEQ "CROSS"
07 X^2
08 X<>Y
09 X^2
10 +
11 X<>Y
12 X^2
13 +
14 RCL 01
15 RCL 04
16 FS? 01
17 CHS
18 +
19 X^2
20 RCL 02
21 RCL 05
22 FS? 01
23 CHS
24 +
25 X^2
26 +
27 RCL 03
28 RCL 06
29 FS? 01
30 CHS
31 +
32 X^2
33 +
34 X<>Y
35 -
36 SQRT
37 STO 00
38 SIGN
39 4.006
40 XEQ "DOT"
41 FS? 01
42 CHS
43 1
44 +
45 ST/ 00
46 RCL 00
47 END
( 75 bytes / SIZE 007 )
| STACK | INPUTS | OUTPUTS |
| X | / | w |
where w is the norm of the 3-vector w = ( wx , wy , wz )
-if CF 01 w = the composition of u and
v
-if SF 01 w = the relative speed ( with respect
to the 2nd reference-frame )
Example: u = ( 0.4 , 0.5 , 0.6 ) v = ( 0.27 , 0.37 , 0.47 )
0.4 STO 01
0.27 STO 04
0.5 STO 02
0.37 STO 05
0.6 STO 03
0.47 STO 06
CF 01 XEQ "MAGW" >>>> w = 0.973249875
SF 01 XEQ "MAGW" >>>> w
= 0.524478980
3°) Doppler Effect & Aberration
-The frequency and the direction of a photon are changed by the relative
motion of the source and the observer.
-The following program performs these conversions.
S = Source S = Source
S\\---> ß
S/ /---> ß
\ \
y
/ /
\ \
|
/ /
cS \ \
c0
| cS/
/ c
ß = velocity of the source , we assume ß
// (Ox)
.
.
| .
.
cS = velocity of a photon, seen in the source reference-frame
. µS
. µ0
| . µS .
µ0
c0 = velocity of the photon, seen in the
observer reference-frame
----------------------------------|----------------------------------
x
O = Observer
µS = the angle ( ß , -cS
) = ( Ox , -cS )
µ0 = the angle ( ß , -c0
) = ( Ox , -c0 ) is measured by the observer
fS = frequency in the source reference-frame
Of course || cS || = || c0 ||
= 1
f0 = frequency measured by the observer
Formulae:
f0 = fS ( 1 - ß2
) -1/2 ( 1 - ß cos µS )
cos µ0 = ( cos µS - ß ) ( 1 - ß
cos µS ) -1
fS = f0 ( 1 - ß2
) -1/2 ( 1 + ß cos µ0 )
cos µS = ( cos µ0 + ß ) ( 1 + ß
cos µ0 ) -1
-We assume that µS and µ0 are
between 0 and 180° and ß is non-negative
-The source is moving away if µ0 < 90°
Data Registers: /
Flag: F00
CF 00 for the conversion ( f0 , µ0 )
>>> ( fS , µS )
SF 00 ------------------ ( fS , µS )
>>> ( f0 , µ0 )
Subroutines: /
01 LBL "DAB"
02 RCL Z
03 FS? 00
04 CHS
05 STO T
06 X^2
07 SIGN
08 ST- L
09 X<> L
10 CHS
11 SQRT
12 /
13 RDN
14 COS
15 ST+ Z
16 *
17 1
18 +
19 ST* Z
20 /
21 ACOS
22 X<>Y
23 END
( 38 bytes / SIZE 000 )
| STACK(CF 00) | INPUTS | OUTPUTS |
| Z | ß | / |
| Y | µ0 | µS |
| X | f0 | fS |
| STACK(SF00) | INPUTS | OUTPUTS |
| Z | ß | / |
| Y | µS | µ0 |
| X | fS | f0 |
Example1: ß = 0.7 µ0 = 41° f0 = 10
CF 00
0.7 ENTER^
41 ENTER^
10 XEQ "DAB" yields
fS = 21.4004 X<>Y µS
= 17.8522°
-Likewise, ß = 0.7 µ0 = 150° f0 = 10 give fS = 5.5141 and µS = 114.9367°
Example2: ß = 0.7 µS = 41° fS = 10
SF 00
0.7 ENTER^
41 ENTER^
10 XEQ "DAB" gives in 1.8 second
f0 = 6.6052 X<>Y µ0
= 83.3397°
-Likewise, ß = 0.7 µS = 150° fS = 10 produce f0 = 22.4915 and µ0 = 167.1555°
Notes:
-The frequencies f0 & fS are equal
iff ( ß = 0 or ß cos2 µ0
+ 2 cos µ0 + ß = 0 )
i-e ( ß = 0 or ß cos2 µS
- 2 cos µS + ß = 0 )
for ß = 0.7 the solution is µ0 = 114.1023165° i-e µS = 65.89768346°
-"DAB" works in all angular modes.
-However, the program listed above should not be used if
µ is very near 0° or 180° because ACOS doesn't
give enough accurate results in that cases.
-For example, with ß = 0.7 and µ0
= 0.01° it yields µS = 0.003969568048°
whereas the correct value is µS = 0.004200840261°
-The following variant uses R-P instead of ACOS , it employs the formulae:
Tan µ0 = Sin µS ( cos
µS - ß ) -1 ( 1 - ß2
) 1/2
Tan µS = Sin µ0 ( cos
µ0 + ß ) -1 ( 1 - ß2
) 1/2
01 LBL "DAB"
02 X<> Z
03 FS? 00
04 CHS
05 STO T
06 X^2
07 SIGN
08 ST- L
09 X<> L
10 CHS
11 SQRT
12 ST/ Z
13 X<>Y
14 SIN
15 ST* Y
16 X<> L
17 COS
18 STO L
19 R^
20 ST* Y
21 ST+ L
22 CLX
23 1
24 ST+ Y
25 RDN
26 ST* Z
27 X<> L
28 R-P
29 X<> Z
30 END
( 53 bytes / SIZE 000 )
| STACK(CF 00) | INPUTS | OUTPUTS |
| Z | ß | / |
| Y | µ0 | µS |
| X | f0 | fS |
| STACK(SF00) | INPUTS | OUTPUTS |
| Z | ß | / |
| Y | µS | µ0 |
| X | fS | f0 |
-Examples 1 and 2 produce the same results, but this time, ß
= 0.7 and µ0 = 0.01° yield
µS = 0.004200840259°
-Moreover, this program also works if -180° < µ
< 0°
-6 bytes may be saved if you replace lines 06 to 11 by ASIN
COS but the execution time becomes 3.4 seconds instead of
2.5 seconds.
References:
[1] Robin M. Green - "Spherical Astronomy" - Cambridge University
Press - ISBN 0-521-31779-7
[2] Stamatia Mavridès - "L'Univers relativiste" - Masson
ISBN 2-225-36080-7 ( in French )
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