Earth's temporal gradient

T

TimeLord

Quantum Scribe
Earth\'s temporal gradient

Hi there. Just wanted to point out the obvious result of relativity's subtle effect on time on earth. As you all know, faster objects have their clocks slowed down, and a rotating object (such as earth) has a range of velocities from rest up to some maximum value, depending on radius. As such, we can define a few things

v = r w
dt*(r) = dt B = dt / sqrt(1 - (v/c)^2) = dt / sqrt(1 - (r w/c)^2)

Sorry the notation is clumsy. We can see then that objects near the surface of the earth travel through time a little faster than those in the center. We also know that a spinning object experiences a centrifugal force. We might suppose that when adjacent spaces pass time at different rates a force is exerted on any objects present. Maybe force is the result of an object seeking the slowest clock pace (or the fastest possible rate through time). It makes sense if we think of dt in terms of some sort of energy level (I know, not properly defined in this context), and an object's clock relative to the rest of the universe becoming slower would be a lower and more stable 'energy' state. We might have to make a new term for this 'energy' or at least a formula to relate it to other forms of energy. Tell me what you think. :D
 
Re: Earth\'s temporal gradient

In general we can say something like

F = -(k)gradient(dt*(x, y, z))

where dt*(x, y, z) is the clock rate at location (x, y, z) and k is a constant.
 
Re: Earth\'s temporal gradient

We can simplify to 1 dimension and go further

F(x) = -(k)gradient(dt*(x)) = -(k)(d(dt*)/dx) = dp/dt
then cross multiply
-(k)(dt)(d(B dt)) = dp(dx)
g = B dt
(B is beta)
-(k/B) g dg = dp dx

Now this appears to be a differential equation (g and dg, or dt and d(dt)). Sadly, this is not my strong point. Can anyone help me with this? /ttiforum/images/graemlins/smile.gif
 
Re: Earth\'s temporal gradient

Actually it appears we can just integrate both sides

-(k/B) g dg = dp dx
K = -(k/B)
K g dg = dp dx
integral(K g dg) = integral(dp dx)
F = dp/dt
E = F dx = (dp dx)/dt
E dt = dp dx
(K/2) g^2 = integral(E dt)
dt* = B dt = g = sqrt((2/K) integral(E dt))

I'm tired so I may have made a mistake in my math. /ttiforum/images/graemlins/smile.gif
 
Re: Earth\'s temporal gradient

TimeLord,

Not too bad. You are on the right track but it has been worked out by Schwarzschild and John Wheeler. Take a look at the Schwarzschild Metric and The Principle of Maximal Aging.

In short it comes down to a freely falling bodyin a gravitational field (which includes an orbiting body) takes a path through spacetime that racks up the greatest amount of time, which happens to be the "shortest" path through spacetime.

The full Schwarzschild Metric is:

dS^2 = (1 - 2m/r)dt^2 - (dr^2/(1 - 2m/r)) - r^2 d(phi)^2

where dS^2" is the spacetime interval between two events A & B. The term "dr" gives you your gradient along the radius from the center of mass in differential form.

Obviously if you're looking for a spacetime interval, time and space (length), "mass" doesn't fit. But in Schwarzschild Geometry there is a conversion factor. For earth the mass works out to .444 cm (2 * .444 = .888 cm). Now all of your units match. They are either time or length.
 
Re: Earth\'s temporal gradient

Bugger. /ttiforum/images/graemlins/yum.gif Didn't know the idea was already conceived, but I shouldn't be surprised. Thanks for the reference, it's interesting stuff. Is there a (known) way to use the Schwartzschild metric for time travel to the past? /ttiforum/images/graemlins/smile.gif
 
Re: Earth\'s temporal gradient

Don;t forget two things...
1) the universe (and thus earth) is expanding with acceleration and is ever increasing speed relative to gravitational center.
2) perception of mind is only one dimension away from time and can be thought of as an encompassing dimension.
 
Re: Earth\'s temporal gradient

Bugger. Didn't know the idea was already conceived
Click to expand...

Which is why I said, "Not too bad." Sure, someone else has already worked through the problem but they too had an epiphany based on looking at the implications of special and general relativity. No one spoon fed you the answer. You worked through a problem and came to the correct conclusion.
 
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