Is time dilation based on the formula for period of a pendulum? The theory Albert Einstein put forward about special relativity mentions a possibility for time dilation, in which he states gravity has a considerable effect on time.
And in high school physics we learnt that the time period of a simple pendulum is given by, 
$$T = 2π\sqrt{\frac{l}{g}}$$
Where $l$, $g$ have their usual meanings.
Well, this describes how the period of oscillation experienced by a simple pendulum depends on the gravitational acceleration present.
My question is whether Einstein proposed his view on time dilation based on a similar phenomenon.
It is worth noticing here that as g tends to zero, time period tends to infinity. This doesn't mean that the actual time is lengthening, but the tangential force on the pendulum decreases, which will ultimately cause the pendulum to stop. But the time goes on, as a dead battery on my wrist watch doesn't imply that the actual time has stopped (not even relative to me)! 
 A: No, the relationship between the period of a pendulum and $g$ is simple Newtonian mechanics and unrelated to special or general relativity. This is discussed in the answers to Time period related to acceleration due to gravity (though I hesitate to link this as that question was not well received).
Time dilation was actually known before Einstein formulated his theory of special relativity. Lorentz published his transformations some time earlier, but their physical significance was not understood. Einstein showed that the transformations arose naturally from his theory of special relativity.
By the time Einstein published his theory of general relativity he understood that time dilation is a result of the geometry of spacetime. This applies to special relativity as well as general relativity. I discuss this in my answer to Is gravitational time dilation different from other forms of time dilation?, though you may find this answer goes into a bit too much detail.
A: Take a look at this pendulum periodicity calculator: http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html.  You can compare the period of each swing of the pendulum under different conditions of gravitational acceleration (little g in your formula).
Notice that as gravity grows stronger (as gravitational acceleration becomes greater), the period of the pendulum becomes shorter, and as gravity grows weaker, the period becomes longer, as you correctly point out.  This is the apparent OPPOSITE of the time dilation effect Einstein predicted in General Relativity.  GR time dilation is an analysis of Proper Time, which depends on the world line between two events and the world line followed by a clock as it "moves" between those two events.
The effect of gravitational acceleration on the period of a swinging pendulum is a purely classical mechanical effect.  The period is measured in Coordinate Time, which is time as measured by an observer who shares the same inertial frame with the pendulum.  Although the period becomes mechanically shorter in Coordinate Time, it becomes longer in Proper Time.
In Special Relativity time dilation is based on the difference between time measured by two clocks in two inertial frames, and likewise has nothing to do with "little g" gravitational acceleration, as that term is used in your formula for the period of a swinging pendulum.
The formula for period of a simple pendulum, and Einstein's formulation of time dilation, are apples and oranges, as the saying goes!
A: I just had the same pendulum time dilation problem. Notice this oscillation does not depend on the mass ($m$) of the pendulum (Einstein in his booklet). Only mass Zero is not possible.
Time Dilation (based on Schwarzschild Metric) 
$$T=2\pi \sqrt{\frac{lo}{g(r)}/ \left(1-\frac{2g(r)r}{c^2}\right)}$$
the pendulum (lo) ticking at $T_0=1 sec$ "say high rate per second of coordinate time", while now coming from fare away (higher gravitational potential), so is getting closer to the big mass $M$ (lower gravitational potential), then the pendulum (l>lo) is ticking with $T$ at "less rate per second of cordinate time" then $T_0$ defines at the beginning. (if $\displaystyle{\frac{2gr}{c^2}=1}$ then T= infinite! So the pedulum is not moving; the clock is standing still?). May be stupid mistake?
Hint: 
Slow ticking pendulum (same model but length l) on earth T>To fast ticking pendulum (same model but length lo) far away from earth. So we have to accept a change of length within a change of coordinates - being local reality of the same model. (To be here or not to be here is the question!)
$$l=lo/{\left(1-\frac{2g(r)r}{c^2}\right)}$$
Conclusion:
We need to take into account that the length of the "same model" changes mechanically within the strong gravitational field. It must be a real Change of a real clock required by GR! Actually we have to take into account the mass of the pendulum ball (m) as well as the mass of the string (l) within a strong G-Field. We have to deal with a Change of a real mechanical clock like the pedulum is one. So if the pendulum lo approaches the big mass (M) the length of the pendulum changes from lo to lo/(1-2gr/c^2)
By the way: what's about sqr(k/m) another real pendulum clock? The time Dilation must hold true as well. I think we will understand this "mechanically" when Quantum Gravity is completing GR. So time Dilation measured by this real clock requires Quantum Gravity embedded into GR. My shot from the hip: Thermodynamic Principles allready complete GR. Quantum Gravity is due to the second law of Nature. May be stupid, but crazy enough to be true.
A: @John, also a caesium clock works thanks to Newtonian mechanics as the atoms fall down under the action of gravity. In general relativity "ALL" clocks have to slow down because of a greater gravitational field not only certain kinds. A pendulum is a clock and according to Einstein it should get slower when gravity increases but this doesn't happen. It is actually a problem for general relativity.  
A: time actually becomes infinity when g becomes zero. for example, assume that you are in space without the gravitation pull now try oscillating the pendulum or keep the pendulum at some point .you will see that the pendulum do not oscillate that means time period becomes infinity or it takes infinite time to reach from one point to another.
