## October 31: Gravitational Redshift

Meaning of Equation

The wavelength of a photon leaving a gravitational field increases.

So today is the last day of the Month of Equations and to end this series, I chose my favourite equation, the gravitational redshift. Before we understand the meaning of gravitational redshift, it is important to know the underlying principle from which this beautiful phenomenon stems out. The gravitational redshift is an outcome of a very deep principle of physics: the equivalence principle. What is that? Let us learn.

In physics, mass is defined in two ways. One is the inertial mass and the other is the gravitational mass. Inertial mass is the measure of an object’s resistance to be accelerated by an applied force. This mass appears in Newton’s second law of motion. On the other hand, the gravitational mass is related to the force by which it attracts other masses around it. This is the mass that appears in Newton’s law of gravitation. According to the principle of equivalence, the gravitational mass is equal to the inertial mass. Confused? Consider this example.

You live on Earth. The Earth exerts a gravitational force upon you. Even if you throw something from a height, it will fall on the ground. The acceleration of the falling object is constant and is equal to 9.8 m/s^2 (g). Actually g is a measure of the gravitational field strength of the Earth. Now suppose you are in a small cabin of a rocket. There are no windows to look out. There is no way you can learn about your surroundings. Just a cabin. Now if that rocket isn’t moving at all, and if you are in deep space with nothing around the rocket, then you will feel weightlessness. Even if the rocket is moving with a constant velocity, you’ll feel weightless and you’ll float in the cabin.

But what will happen when the rocket starts accelerating. If the rocket starts moving with an acceleration that is equal to g, you’ll feel your own weight that is equivalent to that on the Earth. Everything in the cabin will make you feel like you’re on the Earth. If you perform any physics experiment in such a rocket, that is accelerating at 9.8 m/s^2, you’ll get the same result as you get on the Earth. This is the principle of equivalence. Experiments performed in a uniformly accelerating reference frame with acceleration **a** are indistinguishable from the same experiments performed in a non-accelerating reference frame which is situated in a gravitational field where the acceleration of gravity = g = –**a** = intensity of gravity field.

The same thing happens in an elevator. When the elevator goes up (negative acceleration a), your weight increases as the value of effective g increases (g-(-a) = g+a) and vice versa for the elevator going down. If the rope of the elevator is cut, you’ll free fall in the elevator and won’t feel any weight at all. This is what is meant by saying that the gravitational mass is equivalent to the inertial mass.

So now we are in a position to understand gravitational redshift. From the discussion above, it is clear that the laws of physics are same for an accelerating frame and a gravitational frame. Thus any frequency shift which can be shown to arise from acceleration of a radiating source could also be produced by the appropriate gravitational field. If a photon of frequency υ is emitted radially outward from the surface of a gravitational mass M, then the photon energy observed at a distance from the mass will be observed to be lower, or “red shifted”.

In simple terms, the photon has to do some “work” to escape the gravitational field of the star or any other object. This change in energy manifests as change in wavelength or frequency of the photon, since its speed has to remain constant. Thus in escaping the gravitational field, the photon loses some energy and thus its frequency decreases and wavelength increases. This is known as redshift. Note that this redshift is very small and isn’t visible with the naked eye. A simple analogy that Dr. Abhas Mitra uses for this phenomenon: It is like a baby getting tired while crawling up the stairs!

So this concludes the Month of Equations series. It was an amazing experience to learn so many new things and share them with the world. All the equations of the series are available here and under the equations section in the menu. These 31 equations were written by:

Rishabh Nakra (Admin) – pursuing Master’s degree in Physics

Yashika Ghai (Editor) – pursuing PhD in Physics (Theoretical Plasma Physics)

Thank you for being a part of this series! Your reviews are welcomed.