Physics Bootcamp

Subsection 59.8.1 Successes of Einstein's Theory

Einstein's theory had an important early successes. In 1900s there was a long-standing mystery about the discrepancy in the accounting of the precession of the perihelion of mercury which could not be understood from Newton's law of gravitation. Accurate measurements on the path of mercury had shown that the point of the closest approach of Mercury to the Sun, the perihelion, does not occur at the same place each year. Other planets also behave this way, but the effect on mercury is quite pronounced due to its proximity to the Sun. The perihelion of mercury, moves around the Sun each year at a rate of 5,600 arcsec per century. The effects of other planets and non-inertial nature of Earth-based frame accounted for 5,557 arcsec per century. This gave a discrepancy of 43 arcsec per century. When Einstein applied his equations to the problem of the motion of mercury, he was successful in accounting for this discrepancy.

A second more popular success was the confirmation of the bending of light by gravitation. During the 1919 solar eclipse Arthur Eddington and Frank Dyson observed the bending of light from stars in the line of sight of the sun during the eclipse. Although the observation was of dubious quality but arguments for its interpretation based on Einstein's theory were sufficient to convince many physicists at the time. The effect is called gravitational lensing, which has now been observed in several systems. A famous one is the four image of a distant quasar, called the Einstein cross, due to gravitational lensing by a nearby galaxy as shown in Figure 59.8.3. The gravitational lensing happens due to the bending of light from the gravitation of a star or galaxy. The light from a star behind a star can be seen due to the bending of the light from the star behind the star in the fore ground which explains multiple images of the same object.

Subsection 59.8.2 Some Solutions of Einstein's Equations

The solutions of Einstein's equations are usually written in terms of the rule for the infinitesimal interval \(ds\) between two events, say \((t,x,y,z)\) and \((t+dr,x+dx,y+dy,z+dz)\text{,}\) where \(t\) is the time, and \(x\text{,}\) \(y\text{,}\) and \(z\) are the Cartesian coordinates.

1. No Gravity. If there is no gravity, we have a flat spacetime In this case, the square of the interval \(ds\) is
   \begin{equation*} ds^2 = c^2 dt^2 - dx^2 -dy^2 - dz^2,\ \ \ \textrm{(flat spacetime)} \end{equation*}
   where \(c\) is the speed of light. You can write this also in a spherical coordinates \(r, \theta, \phi\text{.}\) We get
   \begin{equation*} ds^2 = c^2 dt^2 - dr^2 -r^2 \left( d\theta^2 + \sin^2\theta d\phi^2 \right).\ \ \ \textrm{(flat spacetime)} \end{equation*}
2. Gravity of a single mass. The space outside a nonrotating mass \(M\) was found by Karl Schwarzschild, and is called the Schwarzschild solution or Schwarzschild metric . It is usually written in the spherical coordinates with the mass \(M\) at the origin.
   \begin{equation} ds^2 = \left( 1 - \frac{r}{r_s}\right) c^2 dt^2 - \left( 1 - \frac{r}{r_s}\right)^{-1} dr^2 - r^2 \left( d\theta^2 +\sin^2\theta d\phi^2 \right),\label{eq-Schwarzschild-metric}\tag{59.8.1} \end{equation}
   where \(r_s\) is called Schwarzschild radius and has the following defintion.
   \begin{equation} r_s = \frac{2G_N M}{c^2}.\label{eq-grav-Schwarz-rad}\tag{59.8.2} \end{equation}
   The Schwarzschild radius is usually very small. For instance, if \(M=M_E\text{,}\) the mass of the Earth, which is about \(6 \times 10^{24}\) kg, we get
   \begin{equation*} r_s[\textrm{Earth}] = \frac{2G_N M_E}{c^2} = \frac{2\times 6.67\times 10^{-11} \times 6 \times 10^{24}}{(3\times 10^8)^2} = 0.009\;\textrm{m}. \end{equation*}
   The Schwarzschild metric given in Eq. (59.8.1) appears to go bad when \(r=0\) and \(r=r_s\text{,}\) called the singularities of the metric. Several physicists later found that the singularity at \(r=r_s\) was the artefact of the choice of coordinate system and was not physical. The singularity at \(r=0\) was real and could not be removed by changing coordinates.