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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)!

David Z
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slhulk
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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.

Community
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John Rennie
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  • You claim that gravitational time dilation has the same origin than inertial time dilation. This is a big misunderstanding of both special and general relativity.

    The invariance of line element is an hypothesis known as "the clock hypothesis". Yet, you don't need it at all in order to show gravitational time dilation in constant gravitational field. By strong equivalence principle, a constant gravitational field is equivalent to a uniformly accelerated observer, and one can easily show using Minkowski diagram in SR that time passes slower for such observers.

     – sure May 01 '15 at 09:26
  • Gravitational time dilation comes from the very presence of gravity, or say, acceleration. This has nothing to do with the symmetric time dilation experienced by inertial observers. – sure May 01 '15 at 09:27
  • @sure: in both SR and GR time dilation is derived from the metric, and the metric is the equation describing the geometry of the spacetime. This applies to all time dilation whether it's due to curvature, motion or acceleration. Have you read the question I linked in my answer? – John Rennie May 01 '15 at 09:34
  • Point is, you shouldn't believe that all frames are put on equals footing even in general relativity. It is therefore dishonest to claim that inertial effects are the same as non inertial ones. It's like claiming that in classical mechanics, fictitious forces are "as real forces" as non fictitious ones. There's a difference, even if they both comes from $F = ma$. Don't be fooled by syntactic similarities. – sure May 01 '15 at 09:36
  • @sure: the whole point of GR is that all coordinate systems are equally valid. For example in flat spacetime we can use accelerating coordinates, in which case the metric is the Rindler metric, and this is a useful way of calculating various effects associated with accelerated motion, like time dilation. – John Rennie May 01 '15 at 09:39
  • All coordinate systems are also equally valid in Newtonian mechanics, don't be fooled, once again, by syntactic similarities. – sure May 01 '15 at 09:40
  • @sure: not true. For example Newton's first law is not true in an accelerating coordinate system. However Einstein's equation is true in all coordinate systems whether accelerating or not. – John Rennie May 01 '15 at 09:41
  • Think that this experiment is conducted using clocks. How can one be sure where the time has actually dilated or it's just an error in the clock due to the gravitational effect?? – slhulk May 01 '15 at 09:43
  • You could formulate Newtonian mechanics coordinate free, see newton-cartan theory. What people call "general covariance" is nothing proper to general relativity. Moreover, even in GR, there's a huge difference between inertial and non inertial frames. There's still somehow an absolute inertial structure as in classical mechanics, so there's still different class of (locally) privileged frames. – sure May 01 '15 at 09:44
  • @slhulk: for a local observer, i.e. the scientist by the pendulum, the period will always be $2\pi\sqrt{l/g}$, where $g$ is the gravitation acceleration felt by the scientist. For an observer watching the experiment from far away outside the gravitational field the period will be $2\pi\sqrt{l/g}$ multiplied by a time dilation factor. – John Rennie May 01 '15 at 09:49
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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!

Ernie
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  • How can one be so sure that the clocks on either inertial frames are not affected by the gravity acting on each frame? – slhulk May 01 '15 at 16:46
  • What type of clocks are used? – slhulk May 01 '15 at 16:46
  • Relativistic time dilation doesn't depend on what type of clock is used. Pendulum clocks ARE affected mechanically by the gravity of their planet and/or the acceleration they undergo. But relativistic time dilation happens regardless and "outside of" such classical Newtonian influences. Basically, gravity in the example is the cause BOTH of Newtonian AND of Relativistic effects. Relativistic time dilation will happen regardless of how gravity mechanically affects the pendulum. – Ernie May 01 '15 at 18:16
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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.

mano4848
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  • yes please. Yes smell(ed) fishy. Im made corrections. Now I think it is fomally correct. I made a mistake while changing the Schwarzschild Metric to get g(r) itroduced the correct way. Minimum the Units are now correct. – mano4848 May 07 '16 at 10:36
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@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.

mfc
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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.

user133677
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