Wednesday, September 2, 2015

General relativity @ 100

November 2015 marks the 100th anniversary of Einstein's theory of general relativity (GR). Actually, he published four papers on the subject in November of 1915. Remarkable. The first two papers lead to the field equations of GR while the other two lay out vital consequences of the theory, including the anomalous precession of the perihelion of Mercury (unexplained since 1859) and the gravitational bending of light. The latter phenomenon was famously confirmed by Eddington's 1919 observation of a total solar eclipse and the shift of apparent position for stars whose light passed very near the sun. Einstein, already famous, became a household name -- an early 20th century super star.

While Einstein's 1905 theory of special relativity dealt with constant velocity frames of reference, GR  dealt with accelerating reference frames. Fond of thought experiments, for special relativity Einstein imagined an elevator in space moving at constant velocity when a horizontal light beam entered from one side. As the light progresses across the small space, the elevator is moving up so that the light appears to exit on the other side below where it entered. But since the velocity is constant, the beam is still straight, just inclined to the floor.

For GR, the same thought experiment gives a different result. Because the elevator is accelerating upward, if we were able to track progress of the light beam across the elevator in equal time intervals the early intervals would show little movement toward the floor, while later ones would have more shift as the elevator speeds up. In other words, the light would appear to bend down toward the floor. Since the only known force that can accelerate everything equally is gravity, this little experiment leads to the idea that light bends in a gravitational field. The bending is small, unless the gravity is due to a very massive body leading to large acceleration.

What connection, you may ask, can GR possibly have to exploration geophysics? Well, there are several. For one thing, the GR field equations are tensor equations as are the field equations of elasticity. Also in both, the equations are so difficult that exact, analytic solutions are few. For elastic, this is basically limited to a point source in unbounded constant velocity media (Stokes, 1849) or near a plane interface (Caniard, 1939) or some ungeologic shape like a sphere or cylinder. For GR, the first (and most useful) is the field around a chargeless, non-rotating spherical object  (Schwarzschild, 1916) which introduced the concept of what we now call a black hole.

But a more interesting similarity resides in the concept of gravitational lensing. This relates to the bending of light by massive objects that lie between us and distant stars. Gravitational lensing can distort the light from stars and galaxies into a bestiary of curiously-shaped (and named) objects: Einstein crosses, rings, arcs and arclets.

While those of us in seismology are at ease with the idea of rays bending in 3D, this kind of distortion is unfamiliar. In the summer of 2004 I visited Stanford University at the invitation of Jon Claerbout. One day I was looking at some gravitational lensing images online when Jon stopped by and mentioned that the same thing must happen with seismic waves in the near surface of the earth. Specifically, the low velocity layer can act as a distorting lens, focussing and defocusing deep reflection energy as it passes upward toward surface receivers. If the effect were big enough we would see it as time shifts that we call statics. But more subtle velocity features would only show up as amplitude anomalies created by focusing and defocusing seismic wavefront energy.

Jon pointed me to pages 154-8 of his wonderful 1985 book Imaging the Earths Interior. In that section he discusses Einar Kjartansson's 1979 PhD thesis, part of which was work along these lines. Einar studied 2D data from the Gulf of Mexico and showed that anomalous pods of material in the earth would show up in the data differently if they were shallow, intermediate or deep; just as gravitational lensing depends on the relative location of source, distorting object, and observer.

That was 1979 and in 2004 Jon looked at me and said that if someone could understand gravitational lensing and bring that to the seismic problem, we would know a lot more about the near surface than we do now. Who knows, maybe he hoped that I would be able to do it. Alas, my abilities fall far short of this formidable problem.

But maybe, just maybe, someone reading this will know enough of both worlds to take on the challenge. I think young Einstein would have jumped on it.

References cited:

Stokes, G.G., 1849. On the dynamical theory of diffraction, Trans. Camb. Phil. Soc., 9, 1-62. 

Cagniard, L., 1962, Reflection and refraction of progressive seismic waves, McGraw-Hill. Translated from Cagniard (1939).   

Schwarzschild, K., 1916,  Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 7: 189-196. (Title translation: About the gravitational field of a mass point in Einstein's theory)

Claerbout, J. F., 1985, Imaging the Earth's Interior, Blackwell Scientific Publications.