A Quick Overview of Some Elementary Aspects of the General Theory of Relativity (GTR)
Hi, Kristy234, and welcome to PF!
Kristy234 said:
I'm confused. If the speed of light is changing
I think you misunderstood Gokul4320's jest (read his post again).
Kristy234 said:
Also, I read somewhere about light being refracted by gravity...but how does this work since it has no mass. My original thoughts were about general relativity and how gravity affects time, so if light is near a massive object, time is slower and therefor it will go slower...?
First, there is a distinction between
infinitesimal velocity (for electromagnetic or gravitational radiation, according to the
general theory of relativity (gtr), the infinitesimal speed of light in a
vacuum is always unity),
velocity in the large (multiple distinct operationally significant notions), and
coordinate speed. In gtr, the coordinate speed of light would be something like dx/dt where x(t) is a null geodesic and where the
time coordinate t need not correspond to
proper time. Such a coordinate speed need not be unity.
However, coordinates are in general not physically significant so this does not contradict the fact that in gtr the speed of light in vacuum is always unity. To take a more familiar example, in a
polar coordinate chart on the
euclidean plane, with
line element
<br />
ds^2 = dr^2 + r^2 \, d\phi^2, \; 0 < r < \infty, \; -\infty < \phi < \infty<br />
the equation of a straight line has the form r = r_0 \, \sec(\phi-\phi_0), and d^2r/d\phi^2 \neq 0, but this does not mean that the line is really "bending"! A
coordinate-free notion of bending is provided by the
covariant derivative of the tangent to a curve.
(In gtr, the covariant derivative taken along a
timelike curve of the
unit tangent vector to said curve is the
acceleration vector whose
magnitude, the
path curvature, tells us whether or not the particle whose world line is represented by this curve is bending, i.e. whether or not our particle feels any acceleration. If not, the pathc curvature vanishes and our timelike curve is a
timelike geodesic.)
Second, all gravitational phenomena are represented in gtr by the curvature of spacetime itself (to be be precise, by the
Riemann curvature tensor), and mathematics tells us that one way in which curvature becomes manifest is that initially parallel
geodesics (the analog, in a curved manifold, of a "straight line") will diverge or converge. This effect is called
geodesic deviation. You can see how it works on the surface of a globe of the world: the longitude lines are geodesic paths, and neighboring longitudes are parallel at the equator but
converge as you move North or South. This convergence is one hallmark of
positive Gaussian curvature; on a "saddle surface", which has
negative Gaussian curvature, initially parallel geodesics will
diverge (you can see this in Escher's Circle Limit woodcuts).
In gtr we use curved
Lorentzian manifolds to model the geometry of spacetime. The spacetime model plays a dual role: not only does it provide the geometric setting for nongravitational physics, but its curvature competely describes all gravitational phenomena. In gtr, the world of a laser pulse (or "photon" if you prefer) traversing a vacuum region is represented by a special kind of geodesic called a
null geodesic. (Because a photon is a massless particle; the world line of a particle with positive mass is a
timelike geodesic if this particle is in a state of
inertial motion.) The appropriate
component of the curvature tensor near a massive object is
negative, which means that in this scenario, initially parallel null geodesics will
diverge. This leads to the so-called
gravitational red shift effect, one of the four
classical solar system tests of gtr (or better say, test comparing
metric theories of gravitation including gtr and various competing theories).
Coming back to "speed of light", there are speculative
variable speed of light theories, but these have not been accepted and so far seem to have no widely accepted experimental support. These theories are talking about something different from the effects we have discussed so far, in the context of our gold standard theory of gravitation, gtr.
As for "refraction", I can only assume that you were misled by this comment:
turbo-1 said:
Einstein claimed that the constancy of the speed of light in a vacuum was confined to a special case (the Special Theory of Relativity) and was invalidated when the gravitational effects of embedded masses needed to be considered. He regarded gravitational lensing as an example of classical refraction, and spent much of the rest of his life trying to determine what properties of space could be modified by embedded matter, and how the variations in these properties affected the propagation of EM through the vacuum.
This is a good example of why ascribing what would now be considered a fringe viewpoint to Einstein entirely out of what is invariably a complex and subtle historical context tends to be (IMO)
perversely unhelpful. Although I am sure turbo meant well, he terribly misled you here, Kristy. My reasons for this judgement are too numerous to list, but let me just say this: even if turbo had accurately described AE's views V(T,C) at time T in their full historical context C--- which he most certainly did
not--- and even if V(T,C) would now be considered flat out wrong--- which is
not true in this case, if one restores the missing context!--- Einstein died in 1955, before the
Golden Age of Relativity (c. 1960-1975), which completely transformed our knowledge and appreciate about gtr and its applications to astronomical observations, so it makes sense to direct newbies to modern textbooks, rather than writings by Einstein which would require vast additional reading in the contemporary physics literature (plus published private correspondence) in order to properly appreciate the context in which his public and private statements must be understood. In addition, it is essential that you understand that Einstein's views (as expressed in his papers and in his private letters) changed frequently and drastically, particularly those pertaining to physical issues. In his thoughtful scientific biography, Abraham Pais writes that this changeable character is in fact one hallmark of Einstein's genius.
Turbo, I hope that in future you will be more careful to avoid the appearance of pushing a fringe viewpoint by presenting a misleading description of some mythical "Einstein's view"
(To mention just one objection, while by an arguably perverse shift in viewpoint one can model weak-field lensing in terms of a kind of "refraction", you seem to have forgotten about
strong field lensing, which is much more complicated, yet described in fully nonlinear gtr exactly the same way--- see Chandrasekhar,
Mathematical Theory of Black Holes.)
Kristy234 said:
Can anyone tell me how this works?
If you want to truly understand all this, you need to study the math. In order to understand gtr you need to have a solid grasp on many subtleties involving curved spacetime geometry, and to grasp these you need to have a solid grasp on various subtleties involving flat spacetime geometry, the kind used in str. A good place to begin is Taylor & Wheeler,
Spacetime Physics. After that you can try the undergraduate level gtr textbook by D'Inverno,
Understanding Einstein's Relativity.
