Proof of curvature: gravitational lensing

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Einstein predicted that the path of light should be deflected by the presence of a massive body. How does this happen? Well, with an enormous amount of matter bending space-time, and this being the medium upon which light propagates, deflection is inevitable.

Above is a simulation of a black hole passing in front of a galaxy in the background. We can see the effect the curvature of space around the black hole has on the background galaxy’s image. Animation courtesy of

A wonderful history of the discovery of gravitational lensing is at the website Here is a link to the specific page, for your convenience.

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Above we see a quite strange image. The really bright objects in the middle are a cluster of galaxies deemed RCS2 032727-132623, which is about 5 billion light years away. The weird blue smears are a galaxy located 10 billion light years from Earth. The light from these galaxies passes 032727, traversing through the warped space-time, and comes to us as a distorted image. So we have a funny smudge, but can these be reconstructed into an actual image of the hidden galaxies?

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The answer, obviously, is yes. The image above shows a reconstruction of what the background galaxy should look like. The physics that describes how space is bent by matter are known. Also, the mass of the intervening galaxies can be estimated. So reconstructing the image of the galaxy being lensed are possible.

Attached is an Optics Paper, written by myself in undergraduate studies, on the topic of gravitational lensing. It is very accessible mathematically with no tensors (I had not yet been exposed to such concepts).


If one knows how much of a depression in space-time an object makes (given it’s mass), and the distance from the object a photon will pass, then its deflection angle is easy to find. The equation detailing the deflection angle of a photon passing by a massive object is listed below (note that this angle is with respect to the original path of motion of the photon):

Θ = 4GM/rc2

Θ in the above equation represents the angle, G is a universal constant for gravitation, M the mass of the object, and r the distance from the object. Of course, our view of space-time as a two-dimensional plane with a dip in it is a bit off.

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If we take into account that space-time is four dimensional, we get an image of curvature more similar to that shown above. The curvature is spherically symmetrical, and I must admit gave me a headache upon first consideration of the above image. The equation for lensing still holds, we just have to apply it in three dimensions to reconstruct images (like the lensed galaxies listed earlier).

Wikipedia has an excellent page on gravitational lensing here.


The Shape of the Universe