Einsteins Twin Paradox
Description
Origin: David M. Weingold in HP-67/HP97 User's Library Solutions - Physics
The program is arranged to calculate subjective and real time differential between an observer on Earth and the pilot of a vehicle accelerating near the speed of light.
If you imagine twins at age 21. One becomes an astronaut and volunteers for the first interstellar flight. He takes off and travels at a ponderous speed of say 2.994444444×108
meters per second. In this situation it is accurate enough to call c the speed of light, 3×108
. The astronaut travels for what he measures to be a year well past the sun at which time he fires retro and navigational engines, and turns around and heads toward Earth; the journey naturally takes another year. He is now 23 years old but when he steps from the ship his twin is over 37 years old! That over 16 years had passed on Earth.
The explanation as to why this happened involves very complicated non Euclidian geometry and relativistic considerations of accelerating frame of reference too complicated for this discussion. It sufficies to say that in the event of tremendous accelerations such as the turning around of a space craft traveling near the speed of light that the order of magnitude of energy involved is extremely large and the consideration of it as it interelates to space as time is conceived as a fourth physical dimension of space, the Universe is then conceived as a giant four dimensional sphere with a three dimensional surface.
The space craft in its turning travels relative to the Earth, not as far along that fourth dimension and hence the differential between the twins age. The equations for this case are quite simple and adequate for this case. They consist primarily of the Lorentz transform i.e., √(1 − v²/c²),where v is the velocity of the space craft relative to the Earth, and c is the universal constant, 3×108
meters per second, the speed of llight. The program inputs consist of speed of space craft in meters per second, time passed on Earth, time passed on board the craft, and the ages of the twins before the flights. With input TE, time passed on Earth, the equation
v2
TS = TE√(1 - ——)
c2
gives TS, time passed on board ship during journey. Input TS, time passed on board ship, and the equation:
TS
TE = ————————————
√(1 − v²/c²)
gives you TE, time passed on Earth during journey.
The label A clears and initiates the program. Label B is the input for average velocity of the craft. Label C is the input for time passed on earth in years and outputs time passed on board ship by hitting GSB 1. Label D input time passed on board ship. GSB 3 give appropriate time passed on earth. GSB 2 and 4 give the ages.
Note:
Be certain that you enter the speed of the space craft in meters per second. All time and age entries must be in years. Outputs will be in years. Do not try to make the space travel at the speed of light, = (3.00x108mete[§/second) as this will only show an error as should be and is implied by the theory of relativity
Sample Problem:
Suppose two twins, age 30, take part in this experiment. The velocity of the ship will average 2.999×108
.
- If the twin on board travels a total of one year how much time will have passed on Earth?
- And given that 25 years passes on earth before return of the ship, how much time passed on board?
- What was the age differential in both cases?
Solution:
- To initiate, clear registers, set display: A →
9.000 16
- Enter ship average velocity in meters/sec: 2.999 EEX 8 B →
8.994 16
- Enter time passed on ship in years : 1 D →
3.155 07
- Calculate time passed on Earth in years: GSB 3 →
3.873 01
- Calculate how old Earth man is upon end: 30 GSB 4 →
6.874 01
- Enter time passed on Earth in years: 25 C →
7.889 08
- Calculate time passed on ship in years: GSB 1 →
6.454 -01
- Find age of Space twin at end of journey: 30 GSB 2 →
3.064 01
Always enter time in years, and speed in meters per second, to go to new case, i.e. new speed or different amounts of time or ages press A, and then continue from step #3 with new values.
Program Resources
Labels
Name |
Description |
|
Name |
Description |
|
A |
Initiate |
|
2 |
Earth twin final age |
|
B |
Set average ship velocity |
|
3 |
Space time to Earth Time |
|
C |
Set time passed on Earth |
|
4 |
Space twin final age |
|
D |
Set time passed in Space |
|
9 |
# Calculate 1 - v²/c² |
|
1 |
Earth time to Space time |
|
|
|
|
Storage Registers
Name |
Description |
|
Name |
Description |
|
0 |
3600 (s/h) |
|
.0 |
c² |
|
1 |
24 (h/d) |
|
.1 |
v² |
|
2 |
365.25 (d/y) |
|
.2 |
Time passed on Earth [s] |
|
3 |
Ts [y] |
|
.3 |
Time passed in Space [s] |
|
4 |
Te [y] |
|
|
|
|
Program
Line |
Display |
Key Sequence |
|
Line |
Display |
Key Sequence |
|
Line |
Display |
Key Sequence |
|
000 |
|
|
|
030 |
42,21,13 |
f LBL C |
|
060 |
42,21, 3 |
f LBL 3 |
|
001 |
42,21,11 |
f LBL A |
|
031 |
45 0 |
RCL 0 |
|
061 |
45 .3 |
RCL . 3 |
|
002 |
42 34 |
f REG |
|
032 |
20 |
× |
|
062 |
32 9 |
GSB 9 |
|
003 |
42, 8, 3 |
f SCI 3 |
|
033 |
45 1 |
RCL 1 |
|
063 |
10 |
÷ |
|
004 |
43 35 |
g CLx |
|
034 |
20 |
× |
|
064 |
45 0 |
RCL 0 |
|
005 |
3 |
3 |
|
035 |
45 2 |
RCL 2 |
|
065 |
10 |
÷ |
|
006 |
6 |
6 |
|
036 |
20 |
× |
|
066 |
45 1 |
RCL 1 |
|
007 |
0 |
0 |
|
037 |
44 .2 |
STO . 2 |
|
067 |
10 |
÷ |
|
008 |
0 |
0 |
|
038 |
43 32 |
g RTN |
|
068 |
45 2 |
RCL 2 |
|
009 |
44 0 |
STO 0 |
|
039 |
42,21, 1 |
f LBL 1 |
|
069 |
10 |
÷ |
|
010 |
2 |
2 |
|
040 |
45 .2 |
RCL . 2 |
|
070 |
44 4 |
STO 4 |
|
011 |
4 |
4 |
|
041 |
32 9 |
GSB 9 |
|
071 |
43 32 |
g RTN |
|
012 |
44 1 |
STO 1 |
|
042 |
20 |
× |
|
072 |
42,21, 9 |
f LBL 9 |
|
013 |
3 |
3 |
|
043 |
45 0 |
RCL 0 |
|
073 |
45 .1 |
RCL . 1 |
|
014 |
6 |
6 |
|
044 |
10 |
÷ |
|
074 |
45 .0 |
RCL . 0 |
|
015 |
5 |
5 |
|
045 |
45 1 |
RCL 1 |
|
075 |
10 |
÷ |
|
016 |
48 |
. |
|
046 |
10 |
÷ |
|
076 |
16 |
CHS |
|
017 |
2 |
2 |
|
047 |
45 2 |
RCL 2 |
|
077 |
1 |
1 |
|
018 |
5 |
5 |
|
048 |
10 |
÷ |
|
078 |
40 |
+ |
|
019 |
44 2 |
STO 2 |
|
049 |
44 3 |
STO 3 |
|
079 |
11 |
√x̅ |
|
020 |
3 |
3 |
|
050 |
43 32 |
g RTN |
|
080 |
43 32 |
g RTN |
|
021 |
26 |
EEX |
|
051 |
42,21,14 |
f LBL D |
|
081 |
42,21, 2 |
f LBL 2 |
|
022 |
8 |
8 |
|
052 |
45 0 |
RCL 0 |
|
082 |
45 3 |
RCL 3 |
|
023 |
43 11 |
g x² |
|
053 |
20 |
× |
|
083 |
40 |
+ |
|
024 |
44 .0 |
STO . 0 |
|
054 |
45 1 |
RCL 1 |
|
084 |
43 32 |
g RTN |
|
025 |
43 32 |
g RTN |
|
055 |
20 |
× |
|
085 |
42,21, 4 |
f LBL 4 |
|
026 |
42,21,12 |
f LBL B |
|
056 |
45 2 |
RCL 2 |
|
086 |
45 4 |
RCL 4 |
|
027 |
43 11 |
g x² |
|
057 |
20 |
× |
|
087 |
40 |
+ |
|
028 |
44 .1 |
STO . 1 |
|
058 |
44 .3 |
STO . 3 |
|
088 |
43 32 |
g RTN |
|
029 |
43 32 |
g RTN |
|
059 |
43 32 |
g RTN |
|
|
|
|
|