Gravitational Lenses

In 1911, Albert Einstein wrote an article about the weight of light. In it, Einstein derives the equivalence of inertial and gravitational mass. It follows that, because of this and because of the mass-energy equivalence E = c² m, gravity must also act on energy. Light, for example, is ultimately energy that travels in portions of E = h f, where f is the frequency of the light and a measure of its colour. Consequently, Einstein concluded, the mass of a body must also ‘attract’ light, since light is energy and can be converted into mass using E / c² = m, which can then be used in Newton’s law of gravity as the second (test) mass. Light packets (photons) do not have inertial mass, but they do have gravitational mass.

In everyday life, this effect is too small to be observed. We see a laser pointer dot hitting the wall at the same height at which we emitted it, because on Earth this deflection is smaller than the radius of an electron.

Do the math:

In 1915, Einstein published the GRT, his own new theory of gravity. It no longer describes gravity with force arrows and fields as in the romantic natural philosophy of the 18th/19th centuries, but as a curvature of space, i.e. with pure geometry. However, this now yields a different result for the strength of this effect of masses on light: The ‘deflection angle’ at which an observer sees a distant object behind a mass is now twice as large as calculated classically.

The deflection angle is calculated using the following formula:

δ = 2 RS a rQ – rL rQ   ,

with Schwarzschild Radius

RS =

2 G M c2

.

Mass M of the Earth: 5.97 * 10²⁴ kg

Radius of the Earth, i.e. distance a of the laser pointer beam from the centre: 6400 m

Speed of light c = 3 * 10⁸ m/s

Gravitational constant G = 6.67 * 10^-11 m³ /s² kg

Distraction

Task: Observe the effects of gravitational lensing in the image below!

A glass lens that deflects light in the same way as a gravitational lens was first built by Sjur Refsdal in Hamburg.

In the local school laboratory ‘Astronomiewerkstatt’, there is a poster that tells the story of the discovery of the gravitational lens effect. Using the portrait of the Potsdam observatory director Karl Schwarzschild, you can study the effects of gravitational lenses on an original image:

… wrote Einstein’s colleagues at the Swiss Patent Office in 1919 during experimental verification of the validity of the general theory of relativity.
In this way, multiple images of the same object can be observed in the sky:

From the formula for the deflection angle, we can see that the greater the mass and the closer the light beam passes by it, the more the light is deflected. With lenses that we are familiar with from opticians, this is usually different.

The middle pictures show a convex and a concave lens: The centre ray simply passes through, and the greater the distance from the centre ray, the greater the deflection.

With a gravitational lens, it is the other way round: the closer the light ray is to the centre, the greater its deflection. That is why the optical (glass) analogue of a gravitational lens is not something you can buy from an optician, but the base of a wine glass (right).

The enveloping curve of the deflected light rays (marked in turquoise on the left for a convex lens) is a line on which a particularly large amount of light arrives because a particularly large number of light rays intersect at each point on this line. If we were to place a screen perpendicular to the image plane, we would see a very bright halo of light, known as the ‘caustic’ (or caustic lines).

Caustics

Please note: When evaluating and analysing gravitational lenses, scientists use ray shooting simulation calculations. This involves shooting rays of light from us into space (as ‘rays of sight’ were thought to be in ancient times) and seeing where they arrive. They are ultimately deflected by our gravitational lenses (masses).

So we set up a virtual ‘screen’ somewhere in space and see where our rays of sight land on this screen.

They land in caustic lines (left in the figure), and from the caustic lines in the simulation, we can infer the lensing mass concentrations in the universe.

Simple Experiment

Caustics: Here we simulate such caustics using the base of a wine glass (which is the optical analogue of a gravitational lens): The caustic lines (shown in turquoise at the top left of the image) appear on the floor as lines of light amplification or collection.

Lecture Experiment

Another Classroom Experiment

We have learned that the optical analogue of a gravitational lens looks like the base of a wine glass. So, for freehand experiments, we find a suitable wine glass, knock off the bowl and play around with it systematically.
For example, you can clamp it between two school desks and point a laser pointer at a wall.
CAUTION – NEVER POINT A LASER POINTER AT PEOPLE!

Some of the most important effects are immediately apparent: distorted multiple images of the laser pointer (black) are visible as reflections in the glass, and the deflection of the light beam is obvious: it hits the lens to the left of the centre and lands far to the right of the lens (behind the neighbouring table).
Here is a closer look at our lens, where this can be seen more clearly: You can also clearly see that the style of the wine glass (ground) is blunt and, as with a real Gravi lens, there is no light from the source in the centre.

Amplification of Light

How big is the light amplification μ in quantitative terms?

Calculate it: The distance between the light beam and the lens is called the impact parameter u. The light amplification depends solely on this parameter:

μ = 2 u2 +2 u √ u2 + 4   , with impact parameter u.

For practical reasons, astronomers measure distances in the sky in Einstein radii. If the light beam passes exactly one Einstein radius past the lens, then u=1.

Calculate: How great is the light amplification?

Consider limit values: Where does the light amplification go for u → 0, i.e. light rays within the Einstein radius?

Does this contradict the law of conservation of energy?

No! To simplify, you can think of it this way:

Normally, only the central ray would reach us. But now two rays reach us, the upper and the lower. Without the lens, these would have gone somewhere else in the universe, i.e. a shadow would now appear somewhere else because the rays are here with us: We can observe this live at the base of our wine glass. The transparent glass casts a shadow in the sunlight that is just as dark as the shadow of my arm (opaque), because all the light that passes through the glass is concentrated in the caustic lines.