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How did Einstein get to the fact that mass slows time?

  1. Sep 30, 2011 #1
    Please correct me if I have made a mistake, but as I know, being closer to large masses will slow the time for you. And I know Einstein thought of that, but my question is... how? How could he get to it by mere thinking? I do understand he had great imagination, but again...
  2. jcsd
  3. Sep 30, 2011 #2


    Staff: Mentor

    Actually, Einstein didn't just think of this all at once, and not all the thinking that led to the conclusion was done by him. Also, there is more than one way to arrive at the conclusion. The full derivation of the prediction, from General Relativity and the Schwarzschild solution, is rather different than the train of thought that Einstein originally followed, and I won't go into it here unless you want me to, since it requires more knowledge of the math.

    The basic argument that Einstein published in 1907 to predict gravitational time dilation is described in this thread:


    It's only fair to tell you that a number of posters in that thread said that the argument was confusing, so reading it may not help you much. :redface:

    Another simple argument one could use is to start with the gravitational redshift, which has been measured experimentally, as described here:


    Einstein predicted this as well, based on the principle of equivalence. This shows that, for example, light that travels upward in a gravitational field must redshift. But a redshift means that the light is losing energy; and since energy is related to the rate of time flow, an observer higher up who receives the redshifted light could conclude that time must be flowing more slowly lower down in the gravitational field where the light came from. However, there are people who object to this argument too. :wink:

    So I guess the best answer to your question is that Einstein came up with a couple of simple arguments that *suggest* that time should slow down as you go deeper into a gravitational field, but he didn't really *prove* it; that took much more theoretical development, and ultimately experimental confirmation, before it was generally accepted by physicists.
  4. Sep 30, 2011 #3
    It was based on his desire to make even accelerating frames "relative" - because of that he developed (or expanded on) the equivalence principle. If that principle is correct, then the same phenomena must be observed in a homogeneous gravitational field (although that hardly exists) as in a constantly accelerated frame. I'm pretty sure that he clearly described that in 1911, but now I can't find that paper so I must tell this from memory (sorry).

    So, light that is sent in the direction of acceleration (thus, "up" inside an accelerating rocket in space) is observed to be Doppler shifted (red shifted) when it is received at the "top".

    From that he predicted that similarly, light that is emitted from lower to higher gravitational potential will also be red shifted.

    Basing himself, as he did for SR, on the wave model of light, he figured out that the number of wave crests in transit cannot change (conservation of cycles).
    For example, if for 1 second a radio wave of 1 kHz is emitted, 1000 cycles have been sent and - at least in vacuum - then it is not possible that less than 1000 cycles arrive. Cycles are also conserved in an accelerating rocket: the 1000 cycles simply arrive over a more than 1 s interval.

    Apparently (I don't recall how he motivated it), he believed that for the equivalent case in a gravitational field the emitter and the receiver are not magically accelerating, without energy input (disclaimer: that's just one of several arguments that I can come up with). The only remaining possibility was that a clock second is longer near heavy masses.

    Thus he predicted that there not only will be redshift, but also gravitational time dilation.
  5. Sep 30, 2011 #4


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    Gravitational time dilation is probably more precisely thought of as space-time curving rather than as "time slowing down". Unfortunately it's hard to describe in lay language why this is true, but I believe it more accurately reflects Einstein's thoughts.

    As to how he realized it, it came out of Special Relativity and calculations regarding the accelerating elevator thought experiment. He realized that there would be doppler shifts when sending signals from the bottom of the elevator to the top, and that these doppler shifts would appear to act just like time dilation.

    At least that's how it's presented in textbooks (sometimes real history is more complex than the simplified textbook variant).
  6. Oct 1, 2011 #5
    Addendum: I suddenly do recall how he motivated it, and it was very different from what I suggested earlier - much more straightforward.

    His desire to create a theory in which even accelerating frames are "relative" implied that someone who appears to be accelerating, may pretend to be actually in rest in a gravitational field. He wanted to describe nature from that perspective.

    Obviously, if an observer assumes to be in rest, the above-mentioned Doppler effect must be zero. His logical conclusion was that what appears as Doppler red shift to an observer who assumes to be accelerating, must appear to an observer who assumes to be in rest in a gravitational field, as red shift due to a different time rate.
  7. Oct 1, 2011 #6


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    Apparently everyone knows a different story how Einstein came up with gravitational time dilation.

    I read one similar to yours based on the EP, but considering light going horizontal (or perpendicular to the frames acceleration). SR predicts that the ray will bend in an accelerated frame. So the light on the inner part of bend moves a shorter distance, than the light on the outer part of the bend. But since the light speed is locally the same everywhere, and the outer part cannot stay behind, the time for the inner part must flow slower.

    I guess he was playing with all those different thought experiments in his head. I also think that rotating reference frames (where resting clocks must tick at different rates based on position, according to SR) played a big role here.
    Last edited: Oct 1, 2011
  8. Oct 4, 2011 #7
    I now verified that indeed, as PeterDonis mentioned, Einstein already figured this out in 1907. Here's a link (regretfully not yet in English, but you can use Google Translate to get an idea):

    His reasoning of 1907 was messier than his improved presentation of 1911 which I discussed above.
    In 1907 he based himself on the fact that in an accelerated frame the speed of light is nearly constant over a small distance. Then, using the definition of simultaneity, he figured out that clocks must appear to run slower in a gravitational field. But I don't fully follow it; if someone else can explain this earlier reasoning, that would be nice!
  9. Oct 4, 2011 #8


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    First he assumes that the Equivalence Principle holds true. Then he shows that clocks at rest in an accelerated frame can get desynchronized according to SR, which based on the initial assumption also applies to clocks at rest in a gravitational field.
  10. Oct 4, 2011 #9
    Ah yes of course - indeed according to the SR synchronization convention the clocks get de-synchronized - and the same must be observed when that is applied in a gravitational field. Thanks! :smile:
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