Sunday, March 17, 2013

Christian Doppler: an anniversary

Irène Joliot-Curie died on March 17th, 1956. She was a member of a high-brow physics family who got her Nobel prize for artificial (alpha-radiation-induced) radioactivity.

Christian Andreas Doppler was born in Salzburg in 1803 and died in Venice 160 years ago, on March 17th, 1853, aged 49. His father, a stonemason, was fortunately observant enough to notice that Christian didn't have sufficient muscles to inherit the craft. Instead, the son went to study philosophy and maths in Salzburg and Vienna.

At the age of 32, Doppler was employed by the Prague Polytechnic – now the Czech Technical University (CTU/ČVUT). He was appointed in 1841. As you can see, when it comes to the affiliation, he was my countrymate. Ethnically, he was Austrian – and "Austria" means the "Eastern Kingdom", i.e. Austrians are effectively Germans who have the maximum experience and credentials in the co-existence with and management of Central and Eastern European nations such as the Slavic ones. I hope that this definition of Austria is fine with the locals. ;-)

The hire was a good idea and the investment was repaid quickly. Just one year later, he gave a talk "On the coloured light of the binary stars and some other stars of the heavens" (in German) to the Royal Bohemian Society. The Doppler effect was postulated in the work and argued to be relevant for the colors of binary stars. As a professor in Prague, he published 50 articles on maths, physics, and astronomy.

In 1847, he left Prague for Banská Štiavnica, Kingdom of Hungary (now Slovakia) to become a professor of mathematics, physics, and mechanics at the Academy of Mines and Forests. In 1849, he moved to Vienna where he became the head of experimental physicists in 1850.

His departure from Prague to a small Slovak town may look anomalous but it was probably made necessary by politics – the revolutionary events of March 1848 – and I guess that Doppler could have been counted as a natural counter-revolutionary. His later physics work in Vienna wasn't important but he became a mentor of a sort for the genetics pioneer Gregor Mendel (who was born, lived, and died near Brno in Moravia, now a part of Czechia).

In 1853, he died in Venice – which also belong to the Austrian Empire at that time: pulmonary disease.

I personally find it puzzling why it took so much time for people to realize that the Doppler effect existed. When Doppler realized those things, seemingly much more complicated wave phenomena had been known for several centuries. Moreover, you can hear the change of the frequency when someone is quickly passing by.

Note that the frequency gets modified to\[

f =\zav{ \frac{1+v_{\rm receiver}/c}{1+v_{\rm source}/c} } f_{\rm orig}

\] The signs of the velocities are chosen in such a way that the frequency increases when the receiver and the source are approaching each other and decreases otherwise.

Note that the receiver and the source play inequivalent roles in the formula; the former enters the numerator while the latter enters the denominator. This is only possible if there is a preferred reference frame – one associated with the medium that carries the waves. That's true for the air that supports sound, for example.

However, special relativity demands that there exists no preferred frame. So only the relative speed \(v_{\rm rel}\) of the receiver and the source must matter. The Doppler formula gets modified to\[

f = \sqrt{ \frac{1+v_{\rm rel}/c}{1-v_{\rm rel}/c} } f_{\rm orig}

\] which is more symmetric. Both the receiver and the source enter both the numerator and the denominator. For \(v_{\rm any}\ll c\), the relativistic and non-relativistic formulae are indistinguishable in the first approximation. Also note that in relativity, the relative speed of two objects is\[

v_{\rm rel} = \frac{ v_{\rm receiver}-v_{\rm source} }{ 1-v_{\rm receiver}v_{\rm source}/c^2}

\] where the signs of the two velocities are measured according to the same convention. All these formulae may be generalized to more general motion (at angles). Also, there is a redshift caused by Hubble's expansion of the Universe whose cause may be translated to the equivalent (for the purpose of calculating one frequency only) motion in the flat space.

The gravitational redshift is a version of the Doppler effect, too. The escape speed is effectively the "right" relative speed between the source and the receiver when they just "seem" to be relatively at rest – the relevant relative equivalent motion (the word "equivalent" is really the same one as the word in the "equivalence principle") we have to substitute arises due to the spacetime curvature.


  1. Hi, Dr. Motl.

    I think your first formula wants a minus sign in the denominator; a typo.

  2. Doppler seems to have been everybody's relative. ;-)

  3. No, it is correct. The only precision to add, is that you have to choose a unit vector for the X-axis, and the correct orientation of this unit vector is from the receiver to the source. For instance, with V_receiver = 0, and the source approaching the receiver, you have V_source < 0, so f > f_orig, which is the correct result.

    And, of course, when V_receiver = V_source, you will have f = f_orig

  4. If my grand grand grand uncle is your relative as well, we must be relative of one another. ;-)

  5. Doppler was also the adviser of Gregor Mendel, the founder of genetics.