Skip to main content

Gravitational redshift

Before I talk about gravitational redshift, let me just briefly talk about doppler shift.

Doppler shift is a high school concept that most of us have learnt at some point or the other in our lives. If you haven’t learnt it, safe to say you’ve surely experienced it. Say you’re on a platform and a train is approaching you. The train is sounding a horn, to catch the attention of passengers on the platform. The increase in the pitch of the horn as the train is approaching you and the decrease in pitch as the train is moving away is the phenomenon of doppler shift.

The same can happen to light. Say a source of light is moving towards you, the frequency you perceive the light to be at is not the same as the frequency of light emitted by the source. This is because of the fact that a clock in the frame of the source ticks slower than your clock, which is at rest. This is the phenomenon of time dilation.

Venturing further now, the Schwarzschild metric describes the spherically symmetric curved space time around a mass. It is the most general vaccum solution. The metric is as follows

Euclidean space in cartesian coordinate system has a metric of the form diag. A matric describes space or spacetime of a manifold, in a specific coordinate system.

Now, let’s say that there is a source at who is emitting radiation and there is an observer at who is receiving the radiation. i.e they are separated radially. Now let’s say that the source emits two pulses of light at and which the observer perceives at and

Given the metric, we can write




Similarly for the second pulse




meaning that for an observer at r = , the time difference between the two events at the source and at the observer are the same.

Now, let’s talk about how clocks at and tick i.e



we can convert the time difference to frequency and therefore energy, therefore changing the above equation to

So, for the case , it can be seen that i.e for a photon emitted at , by the time it reaches , it would have lost a certain amount of energy.

I would like to remind you again that this is curved spacetime and not flat spacetime. In qualitative terms, in a central gravitational potential, a photon will lose energy as it climbs out of the potential.

An interesting nugget of information is the physical implication of clocks in curved spacetime. The spacetime around earth is curved and clocks in space i.e on satellites run a bit slower than a clock on earth. And this difference is crucial to calibrate GPS signals, without which the accuracy with which one can pin point their location becomes abysmal.

This article was written using StackEdit1.


  1. StackEdit is a full-featured, open-source Markdown editor based on PageDown, the Markdown library used by Stack Overflow and the other Stack Exchange sites.

Popular posts from this blog

Animation using GNUPlot

Animation using GNUPlotI've been trying to create an animation depicting a quasar spectrum moving across the 5 SDSS pass bands with respect to redshift. It is important to visualise what emission lines are moving in and out of bands to be able to understand the color-redshift plots and the changes in it.
I've tried doing this using the animate function in matplotlib, python but i wasn't able to make it work - meaning i worked on it for a couple of days and then i gave up, not having found solutions for my problems on the internet.
And then i came across this site, where the gunn-peterson trough and the lyman alpha forest have been depicted - in a beautiful manner. And this got me interested in using js and d3 to do the animations and make it dynamic - using sliders etc.
In the meanwhile, i thought i'd look up and see if there was a way to create animations in gnuplot and whoopdedoo, what do i find but nirvana!

In the image, you see 5 static curves and one dynam…

Pandas download statistics, PyPI and Google BigQuery - Daily downloads and downloads by latest version

Inspired by this blog post : https://langui.sh/2016/12/09/data-driven-decisions/, I wanted to play around with Google BigQuery myself. And the blog post is pretty awesome because it has sample queries. I mix and matched the examples mentioned on the blog post, intent on answering two questions - 
1. How many people download the Pandas library on a daily basis? Actually, if you think about it, it's more of a question of how many times was the pandas library downloaded in a single day, because the same person could've downloaded multiple times. Or a bot could've.
This was just a fun first query/question.
2. What is the adoption rate of different versions of the Pandas library? You might have come across similar graphs which show the adoption rate of various versions of Windows.
Answering this question is actually important because the developers should have an idea of what the most popular versions are, see whether or not users are adopting new features/changes they provide…

Adaptive step size Runge-Kutta method

I am still trying to implement an adaptive step size RK routine. So far, I've been able to implement the step-halving method but not the RK-Fehlberg. I am not able to figure out how to increase the step size after reducing it initially.

To give some background on the topic, Runge-Kutta methods are used to solve ordinary differential equations, of any order. For example, in a first order differential equation, it uses the derivative of the function to predict what the function value at the next step should be. Euler's method is a rudimentary implementation of RK. Adaptive step size RK is changing the step size depending on how fastly or slowly the function is changing. If a function is rapidly rising or falling, it is in a region that we should sample carefully and therefore, we reduce the step size and if the rate of change of the function is small, we can increase the step size. I've been able to implement a way to reduce the step size depending on the rate of change of …