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General Relativity passes the Ratio's Test

Information about the laws of physics is effectively baked into gravitational waves, the ripples in spacetime created when massive objects such as black holes spiral into one another.
Information about the laws of physics is effectively baked into gravitational waves, the ripples in spacetime created when massive objects such as black holes spiral into one another.
(Image: © Shutterstock)

At least 3,700 years ago, Babylonian mathematicians approximated the ratio of a circle's circumference to its diameter. They inscribed their answer, the first discovered value of pi, on a humble clay tablet: 25/8, or 3.125. Now Carl-Johan Haster, a theoretical astrophysicist at the Massachusetts Institute of Technology, has managed to do almost as well: in a study uploaded to the preprint server, he measured pi to be about 3.115.

In the intervening years, researchers have calculated the true value of the ratio to a modest 50 trillion decimal places with the aid of powerful computers (you probably know how it starts: 3.141592653 … and on into infinity). Haster's approximation of it may be a couple of millennia behind in terms of accuracy, but that fact is of little relevance to his real goal: testing Einstein's general theory of relativity, which links gravity with the dynamics of space and time.

Information about the laws of physics is effectively baked into gravitational waves, the ripples in spacetime created when massive objects such as black holes spiral into one another. Haster, a member of the Laser Interferometer Gravitational-Wave Observatory (LIGO) Scientific Collaboration, noticed pi appeared in several terms of an equation describing the waves' propagation.

"What Carl did was say, 'Look, all of these coefficients depend on pi. So let's change pi, and let's check whether the measurements are consistent [with general relativity],'" says Emanuele Berti, a theoretical physicist at Johns Hopkins University, who was not involved in the new study and is not part of the LIGO collaboration.

Haster realized that he could treat pi as a variable instead of a constant. Then he could check the equation for gravitational waves against LIGO's experimental measurements of them. Einstein's theory should have matched the measurements if and only if Haster used values of pi close to that already determined by other methods. If general relativity matched LIGO's measurements when pi was not close to its true figure, that would be a sign that the theory was only half-baked. By trying values of pi from –20 to 20, Haster checked more than 20 observed candidate gravitational-wave events and found that the figure that matched theory to experiment was about 3.115. So Einstein's recipe does not seem to need any tweaking just yet. "In my head, at least, [the study] has a nice mix of being both kind of cute and amusing and also actually producing a valid and fairly strong test of general relativity," Haster says.

Pi seems to pop up all the time—not just explicitly in circles but in the hydrogen atom and the way needles fall across lines. The reason a factor of pi appears in an equation for gravitational waves is a little headier, however: the waves interact with themselves.

"When a gravitational wave is traveling out, it sees the curvature of spacetime, including the energy that was generated by the gravitational waves produced in the past," Berti says. The first stone you drop into a calm pond sends out smooth ripples across the surface. If you drop another stone immediately after, the surface is no longer smooth—leftover ripples from the previous stone will interfere with new ripples from the second one. Gravitational waves work similarly, but the medium is spacetime itself, not water.

The equation describing this self-interacting effect contains factors of pi as a piece of several numerical terms. A previous examination of Einstein's theory by LIGO in 2016 varied individual terms instead of slicing out a common factor across several terms such as pi. Although this approach sufficed as a test of general relativity, physicists have wanted to see all the terms changing together, and Haster's method using pi offers a way of doing just that.

But it remains a far from transcendental test of the theory. One issue is the relative uncertainty of Haster's figures: His approximation of pi currently ranges from 3.027 to 3.163. Significantly sharpening it will require observing mergers of lighter objects such as neutron stars, which create drawn-out gravitational waves that can last 300 times longer than those from a colliding pair of massive black holes. Like trying to identify an unknown song, the more one can listen, the better. Currently, there are only two recorded confirmed neutron star mergers in the available data. And until LIGO—which is shut down because of COVID-19—resumes operations, that number will not change.

Not everyone is worried about the flakiness of this pi-scrying technique, though. "Many people have been discussing the fact that we could maybe change Pi Day (March 14) into 'Pi Two Weeks' (March 2 to March 15) to account for current uncertainty," jokes Chris Berry, an astrophysicist at Northwestern University, who was not involved in the new study and is part of the LIGO collaboration.

This proposal would, of course, likely increase the number of pastries for a pi-loving physicist to consume. But Berry maintains that calorie increase would not be altogether a bad thing. A fortnight of feasting, he says, would eventually give researchers another way to approximate pi: measuring their own rotund circumference.

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  • Xinhang Shen
    It seems the author and the researcher still don't know that Einstein's relativity has already been disproved both theoretically and experimentally.

    The fatal error of relativity is the redefinition of time which is no longer the time measured with physical clocks, as shown in the following:

    We know physical time T has a relationship with the theoretical time t of both Newtonian mechanics and Einstein's special relativity: T = tf/k where f is the frequency of the clock and k is a calibration constant.

    In Newtonian mechanics, since the theoretical time t is the absolute Galilean time and thus the frequency f is a frame independent constant. We can set k = f to make T = tf/k = tf/f = t, which proves our physical time T is the absolute theoretical Galilean time t, and also confirms that our physical time is absolute too.

    Now let’s look at special relativity. We would like to use the simultaneity of events measured with both physical time T and relativistic time t in different inertial reference frame through Lorentz Transformation to verify whether they have the same property.

    If you have a clock (clock 1) with you and watch my clock (clock 2) in motion and both clocks are set to be synchronized to show the same physical time T relative to your inertial reference frame, you will see your clock time: T1 = tf1/k1 = T and my clock time: T2 = tf2/k2 = T, where t is relativistic time of your frame, f1 and f2 are the frequencies of clock 1 and clock 2 respectively observed in your inertial reference frame, k1 and k2 are calibration constants of the clocks. The two events (Clock1, T1=T, x1=0, y1=0, z1=0, t1=t) and (Clock2, T2=T, x2=vt, y2=0, z2=0, t2=t) are simultaneous measured with both relativistic time t and clock time T in your reference frame. When these two clocks are observed by me in the moving inertial reference frame, according to special relativity, we can use Lorentz Transformation to get the events in my frame (x', y', z', t'): (clock1, T1', x1'=-vt1', y1'=0, z1'=0, t1') and (clock2, T2', x2'=0, y2'=0, z2'=0, t2'), where

    t1' = r(t1-vx1/c^2) = r(t-0) = rt
    t2' = r(t2-vx2/c^2) = r(t-tv^2/c^2) = rt/r^2 = t/r
    T1' = t1'f1'/k1 = (rt)(f1/r)/k1 = tf1/k1 = T1 = T
    T2' = t2'f2'/k2 = (t/r)(rf2)/k2 = tf2/k2 = T2 = T

    in which r = 1/sqrt(1-v^2/c^2).

    That is, no matter observed from which inertial reference frame, the two events are still simultaneous measured with physical time T i.e. the two clocks are always synchronized measured with clock time T i.e. clock time T is absolute, but not synchronized measured with relativistic time t'. In real observations, we can only see clock time T but not relativistic time. Therefore, clock time is our physical time and absolute, totally different from relativistic time in Lorentz Transformation and thus relativistic time is a fake time without physical meaning. The change of the reference frame only makes changes of the relativistic time from t to t' and the relativistic frequency from f to f', which cancel each other in the formula: T= tf/k to make the physical time T unchanged. This proves that even in special relativity our physical time is still absolute. Therefore, special relativity based on the fake relativistic time is wrong.

    For more details, you can read the journal paper here:
  • bolide
    In the Buffon's Needle exhibit that the article links to, I don't get why the range of ϴ is from 0 to π. If the needle is horizontal, ϴ = 0 and sin (ϴ) = 0. Rotate the needle 90° ccw, then ϴ = 90°, sin (ϴ) = 1, and (½)sin ϴ = ½. So far, so good. Rotate another 90°, and the line is horizontal again, and sin (ϴ) = 0 again. But there the value of ϴ would be 180°. So where does π come from? Or, in what units is this measured as π? And in that case, in what square units is the area of the rectangle π/2?