>At 12:47 AM 10/2/98 -0700, Dario A Giraldo wrote:
>>But to answer your question on scientific support, I have began with the
>>following formula:
>>
>>(T2 - T1) = (t2-t1)[1 - (V2/C2)]0.5 Relativistic Time given by A.Einstein
>>
>>A year in Jupiter isnāt the same as a year on Earth, if one defines a year
>>as one complete rotation of a planet around the sun. So we can be on
>>Jupiter speaking about a year while on earth time we are talking about tens
>>of years.
It looks here like Dario might have confused a special relativistic time
dilation with differences in the orbital periods of Jupiter and Earth.
However although Glenn's calculation, below, demonstrates the utter
insignificance of such relativistic effects for Earth-bound and Jupiter-
bound observers, it is in error in that it gets the actual amounts of
time dilational effect completely wrong--except that the incorrectly
calculated dilations -- like the corrects ones -- are completely
negligible. Glenn's calculation even gets the sign wrong (predicting
that time on Earth runs slower than on Jupiter) for the relative time
dilation between Jupiter and Earth.
>Let us define the year as the tick on a personal watch of
>
>365.25 x 24 x 60 x 60= 31557600 seconds. So, you take your wrist watch and
>go to Jupiter and ride along with it and I stay on earth (I don't like
>breathing methane, so I will let you do that).
>
>Jupiter is actually traveling slower than the earth (although by a
>miniscule amount). Jupiter travels at 13.1 km/s while the earth travels at
>29.8 km/s. the difference in velocity is 16.7 km/s This is the velocity
>which will be used to calulate the time dilation since on jupter you won't
>be able to tell your own motion so all you need for the above is the
>difference.Given that c= 299792 km/s
>
>Squaring all these values we have
>
>^2=278
>c^2=8.9875 x 10^10
>
>So when you on jupiter look at earth you will see time moving slower
>because the earth is going faster.
>
>you measure
>31557600 seconds on your watch but on earth you see my watch has ticked:
>31557600 seconds(1-v^2/c^2)= T2-T1=earth ticks=
>31557600 seconds(1-278/8.9875 x 10^10)=
>31557600 seconds (1-3.1 x 10^-9)= 31557600(.9999999969)=31557599.90217
>seconds have gone by on earth time.
>
>This is not enough to sneeze at literally less than 1 tenth of a second
>difference in an entire year. How does this solve your problem?
The biggest error in the above calculation is that it completely ignores
the *general* relativistic gravitational time dilation for the Earth and
Jupiter in their respective positions in the Sun's gravitational
potential well. This is a significant error in that the amount of
gravitational time dilation is of the same order of magnitude (actually,
it is precisely *twice* as large for circular orbits) as the special
relativistic velocity-induced time dilation that Glenn has included.
I suppose it may have been reasonable for Glenn to ignore the
gravitational time dilational effects since Dario didn't mention them in
his post, but this neglect makes for significant errors in the
calculated time dilations--although the so-calculated ones are,
correctly, utterly negligible in the end. Other significant sources of
time dilation (esp. for Jupiter) are the gravitational and velocity-
induced time dilations due to the planet's own gravity and the rotational
spin of the planet itself.
Let Newton's universal gravitational constant be G; let the mass of the
Sun be M; let the orbital speed of the planet around the Sun (assuming
a circular orbit for convenience) be V; let R be the radius of the
planet's orbit around the Sun; let the mass of the planet be m; let the
equatorial rotational speed of the planet's surface be v; and let the
equatorial radius of the planet be r. If we (for convenience) assume
that the observers 'on' (quotes here since a gas-ball planet like Jupiter
has no soilid surface to be *on*) the planets are on the planets'
equators then to leading order in the small parameters (i.e. leading
order in 1/c^2) the differential relative time dilation (delta t)/t
(relative to an observer at rest relative to the Sun and located far
beyond the Sun's gravitational pull -- but not so far as to be close to
some other star) is given as:
(delta t)/t = V^2/(2*c^2) + G*M/(R*c^2) + G*m/(r*c^2) + v^2/(2*c^2) .
For a circular orbit the (general relativistic) gravitational time
dilation in the second term is precisely twice as large as the special
relativistic velocity-induced time dilation in the first term. This is
simply a consequence of the relationship (i.e. V^2 = G*M/R) between
orbital radius and orbital speed given by Kepler's 3rd law and the Virial
theorem. Combining the first 2 terms yields:
(delta t)/t = (3*V^2)/(2*c^2) + G*m/(r*c^2) + v^2/(2*c^2) .
Applying this formula to Earth gives:
(delta t)/t = 1.4806 x 10^(-8) + 6.9535 x 10^(-10) + 1.2035 x 10^(-12) =
= 1.5502 x 10^(-8) .
Applying this formula to Jupiter gives:
(delta t)/t = 2.8457 x 10^(-9) + 1.9717 x 10^(-8) + 8.7931 x 10^(-10) =
= 2.3442 x 10^(-8) .
Notice that in the case of Earth the greatest contribution to the total
time dilation comes from the combined contributions of the Sun's gravity
and Earth's orbit around the Sun (first term), but in the case of Jupiter
the greatest contribution is due to the self-gravity of Jupiter itself
(middle term) with the Sun's contribution being significantly smaller.
In both cases the effect of the motion from the planet's rotational spin
is the smallest contribution, although Jupiter's large rotation rate
combined with its great radius serve to make even this term contribute
almost 4% of the total time dilation. Note also that *Jupiter*, in spite
of its slower orbital speed and greater distance from the Sun, has the
greatest time dilation, *not Earth*. The difference in these time
dilations is 7.940 x 10^(-9). This means that a year of 31557600 seconds
on Earth corresponds to a time interval on Jupiter of 0.251 sec less
than this amount. (Glenn had the time interval on Earth being 0.098 sec
less than the year's time interval on Jupiter.) It should be noted that
my calculation here neglects the further time dilations due to the
orbital motion of the Sun around the Milky Way Galaxy, the gravitational
time dilation from the galaxy's gravitational field, and the overall
motion (with respect to a frame in which the Cosmic Background Radiation
is isotropic) of the galaxy towards the so-called 'Great Attractor'.
The neglect of these extra-solar system contributions is justified as
they are the same for both Jupiter and Earth and their effects cancel out
when comparing corresponding time intervals on Earth and on Jupiter.
>The difference between the year you measure on your watch and the one I
>measure on mine can't even be measured on a personal watch. They aren't
>accurate enough.
Even though the Glenn's calculation is wrong his conclusion/observation
about the utter negligibility of the relative time dilations between
Jupiter and Earth is right on regarding Dario's ideas.
David Bowman
dbowman@georgetowncollege.edu