I just had the same pendulum time dilation problem. Notice this oscillation does not depend on the mass ($m$) of the pendulum (<PERSON> 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. 483
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. 651