Einstein Rings

Einstein was discussing some problems with me in his study when he suddenly interrupted his explanation and handed me a cable from the windowsill with the words, "This may interest you." It was the news from Eddington confirming the deviation of light rays near the sun that had been observed during the eclipse. I exclaimed enthusiastically, "How wonderful, this is almost what you calculated." He was quite unperturbed. "I knew that the theory was correct Did you doubt it?" When I said, "Of course not, but what would you have said if there had not been such a confirmation?" he retorted, "Then 1 would have to be sorry for dear God. The theory is correct."

— Use Rosenthal-Schneider

1 "Did you doubt it?"

Arthur Eddington's verification of the deflection of starlight by Sun in 1919 made Albert Einstein an instant celebrity. In this project we reproduce Einstein's prediction (though not by his method) and apply it to important modern astronomical observational techniques. Einstein's results are correct for deflection of starlight by most spherical astronomical objects and for light passing a Schwarzschild black hole at a large radial distance such that r » 2M. This is called the weak field approximation. Section 7 uses the weak field approximation to describe Einstein rings such as those displayed in Figure 14, page 5-25. The lens-like concentration of light can also increase the amount of light received from a distant star, an effect called microlensing, described in Section 8. In Section 9 we use Einstein rings to account for some features of the image of Saturn on the cover of this book.

Light passing close to a black hole does not satisfy the weak field approximation but is radically deflected and can even go into temporary orbit (Chapter 5). Such radical deflection accounts for the "diamond necklace" in the center of the upper image on the cover, as described in Section 10. (For even wilder behavior of light see Project F, The Spinning Black Hole.)

2 Newtonian Deflection of Light

Before carrying out the analysis of general relativity, begin with an approximate Newtonian prediction of light deflection under the assump-

tion that light accelerates downward in a gravitational field (Figure 1). We make the approximations (1) light is deflected only during its passage across the diameter of Sun, and (2) during this deflection the acceleration is perpendicular to the direction of motion of light and equal in magnitude to the acceleration at the surface of Sun.

QUERY 1 Gravitational acceleration at the surface of Sun. Using physical constants from inside the back cover, calculate the acceleration of gravity at the surface of Sun, according to Newton, in conventional units meters/second2.

Path of light from star

Assune that ail deflection takes place aJong thte segment, with downward acceleration equal to that at radus ft

Path of light toward Earth

Figure 1 Newtonian analysis of light deflection by Sun. Assume that light, along with material particles, undergoes gravitational acceleration. The change in transverse velocity, according to Newton, is reckoned using the acceleration at the surface of Sun acting during the time the light crosses the diameter of Sun (calculus proof I), The deflection, less than two seconds of arc, is too small to be visible in this or later figures, which therefore will show this deflection greatly exaggerated

Apparent direction of star

Distant star

3 x 108 meters) second

Light

3 x 108 meters) second

Figure 2 Schematic diagram (greatly exaggerated) of the deflection of starlight in the Newtonian analysis outlined in Figure 1.

Approx. 1300 meters/second

Figure 2 Schematic diagram (greatly exaggerated) of the deflection of starlight in the Newtonian analysis outlined in Figure 1.

QUERY 2 Time for light to cross Sun. Calculate the length of time in seconds it takes light to move a distance equal to the diameter of Sun.

QUERY 3 Change in "sunward" motion of light. Assume that light grazing Sun experiences a constant acceleration perpendicular to its path equal to that calculated in Query 1 for a time calculated in Query 2 (and no acceleration anywhere else along its path). Then the light picks up a "sunward" component of velocity equal to approximately 1300 meters/second. See Figure 2. Find the result to four-digit accuracy.

QUERY 4 "Newtonian" angle of deflection. Let A<}> be the angle of deflection of light in radians, as shown in Figure 2. Because of the small angle of deflection, assume that tan(A<|>)» A<f>. From the results of the Receding queries, this deflection has the approximate value A<|> «= 4 x 10 radian. Find the result to three-digit accuracy.

QUERY 5 Seconds of arc. The deflection of starlight by Sun is usually expressed in seconds of arc. There are 60 minutes of arc in one degree and 60 seconds of arc in one minute of arc. The deflection A<|> of light by Sun is approximately 1 second of arc—according to this Newtonian analysis. Find the result to three-digit accuracy.

From the Newtonian result (too small by a factor of 2, as we shall see), we draw the provisional conclusion that the deflection is very small. This assumption is important in the general relativistic derivation that follows, and must be verified again for the results of that derivation.

3 Setting Up the General-Relativistic Analysis

Now we move on to the general-relativistic prediction of light deflection. A pulse of light approaches, passes, and recedes from Sun, its position tracked by the azimuthal angle <j> (Figure 3). If there is no deflection, the angle <|> sweeps through n radians as the pulse moves from distant approach to distant recession. We want to find the additional angle A<(> caused by gravitational deflection (Figure 4).

How much, d<}>, does the tracking angle <(> change for each small change in radius dr (Figure 3)? The answer is embodied in the expression d<\>/dr, derivable from equations [15] and [14], page 5-8:

Here b is the impact parameter (defined in Figure 2, page 5-6).

Figure 3 Measuring the change cfy in azimuthal angle $a$a tight puke changes radius dr The deflection is greatly exaggerated if there is no deflection the angle + will sum to n as r goes from the cBstantstar to R and out to distant Earth. To predict the actual deflection we need a relation between dr and cfy Equation [4] gives this relation The cSstance bis the impact parameter

Figure 3 Measuring the change cfy in azimuthal angle $a$a tight puke changes radius dr The deflection is greatly exaggerated if there is no deflection the angle + will sum to n as r goes from the cBstantstar to R and out to distant Earth. To predict the actual deflection we need a relation between dr and cfy Equation [4] gives this relation The cSstance bis the impact parameter

Gravitational Deflection
Figure 4 Gravitational deflection angle A4> of starùght by Sun—greatly exaggerated

QUERY 6 Angle ty changes as r changes. Divide corresponding sides of equations [1] and [2] and simplify to show that ffl=

Show that this can be modified to yield

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