How do the general theory of relativity, the special theory of relativity, and quantum mechanics all fit together?
This question is best answered with an experiment. If you try to answer it with thought experiments, you will make a mistake.
Imagine that you have a pulsed laser beam connecting two, distant points on the Earth with femtosecond precision and accuracy.
Running alongside this laser beam is a pulsed electron beam and you measure the arrival-time of this electron beam relative to the laser beam with femtosecond accuracy using, for example, the optical synchronization system at the European XFEL (many years ago, I wrote my thesis on this).
When the moon pulls on the Earth’s crust, the laser beam system maintains a constant length and you measure changes in the arrival time of the electron beam relative to the laser beam. This represents a change in the distance the electrons traveled through the Earth.
Now, you change the momentum of the electron beam and notice that it also changes the apparent distance traveled through the Earth, but changing the momentum of the electron beam clearly didn’t change the Earth’s radius or gravitational pull, so how should we interpret this?
When the Earth’s radius increases, the Earth gets heavier. When the electron beam energy increases, it gets heavier.
There is a gravitational attraction between the relativistic electrons and the Earth and if we equate the local changes in gravity to the changes in the relativistic mass of the electrons, then g + delta g = (M+delta m) G/r² .
G=(g+delta g)/(M+delta m) * r² might be a new way to measure the gravitational constant, a value which has significant uncertainty of 10^-3
Invited Review Article: Measurements of the Newtonian constant of gravitation, G
The method I describe is comparable to those that measure G in a direction perpendicular to the surface of the Earth rather than parallel to the surface of the Earth.
J. P. Schwarz, Science 282, 2230–2234 (1998).
The sorts of gravitometers which are used in high-precision metrology have a resolution of about 2 micro Gal, while the measurement I’ve described has a resolution which is 1000 times better than this. A single, tidal oscillation measured by the XFEL optical synchronization system is
delta g = 75 +- 0.002 micro Gal
Such an experiment exposes the unusual semantics of special and general relativity. Whenever the Earth’s mass changes locally, the relativistic mass of the electrons has to be changed in direct proportion in order to keep the arrival-time of the electrons constant relative to the optical reference. If the optical reference is tuned to compensate for the change in the local gravitation, then the energy of the electrons will not be as stable relative to a more absolute reference provided by, for example, an atom impinged upon by synchrotron light from the electrons. This is an interesting situation because you have a trade-off between energy stability and arrival-time stability constrained by the uncertainty principle within a large scale accelerator and within a tiny atom.
If the equivalence principle holds, then mg = GmM/r², but in the instant that the electron beam’s energy increases, this equivalence principle is violated. That is why it is important to remember that general relativity is a static approximation of a more dynamic system.
When trying to capture the dynamic aspect of such a system, you need the correspondence principle instead. It says v dm/dt+mg=GMm/r² including a relativistic correction because F=ma is only true for low energy masses. Normally, the test mass cancels out and that is why all masses fall with the same acceleration — unless they are relativistic.
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How does this experiment compare to a gravitational wave experiment like LIGO? One would think that LIGO could measure optical path lengths with greater accuracy than the point-to-point optical synchronization system in the XFEL because LIGO measures one micron optical path-length changes with a precision of one part in 10^-21, after all. But this is not the case. LIGO only measures changes of one part in 10^-21 within a bandwidth of 10 Hz- 1000 Hz because the interstellar, cosmic, gravitational waves in which they are interested have relatively short wavelengths compared to the tidal, gravitational waves produced by the moon. Below 10 Hz, their apparatus has no accuracy whatsoever, while the femtosecond optical synchronization system at the European XFEL only works below 10 Hz and above 10000 Hz.
Calibration uncertainty for Advanced LIGO’s first and second observing runs
LIGO isn’t the only game in town, of course. Other groups use multiple, continuous wave laser sources in interferometry to measure arrival-time changes over relatively short distances. They can calibrate out humidity and temperature effects, but they can’t use this continuous wave technique over kilometers because of scattering in the fibers and because of discrepancies that build up in phase and group velocity when the power is too high.
In addition to the advantage of being able to send higher power signals without distortion, pulsed arrival time or interference measurement of optical signals instantiates the Fellgett advantage over a continuous wave interference measurement because a broadband source is multiplexed by default.
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Referring back to the original question: what is the difference between general and special relativity? The Sagnac loop tells us that light travels a shorter path in a circle when the circle rotates, but the Michelson interferometer tells us that light doesn’t travel a shorter path when it travels in a straight line that is mounted on a circle that rotates — unless the gravity or area of the circle is changing. The reason for this is that two points do not define a space, but three or more points do and contraction or expansion of light must be tied to a volume. The gravitational constant is given in units of volumetric expansion after all. This is why the distance between two points on the Earth, as measured with a laser, may be held constant, while the distance, as experienced by particles under the influence of gravity, may change as the shape of the planet changes.
Subtleties are important in this business it is important to note that volumetric expansion can also be expressed as a localized vibration as when an object sinks within vibrating sand. This heuristic is not described by special or general relativity, but it is inherent in quantum mechanics. Joining the two formalisms requires an understanding of this.
Originally published at http://quora.com.
