Redshift

[Nick James]

The article on redshift currently on the BAA front page gets a bit confused since it uses a non-relativistic definition of Doppler and then has problems explaining redshifts greater than 1.

The simple definition of redshift, z, is v/c where v is the radial velocity and c is the speed of light. This is OK for very slow objects but it gets very wrong for objects with a velocity which is significant relative to the speed of light. The correct definition from special relativity is z = sqrt((1+v/c)/(1-v/c)) – 1. This approximates to v/c if v is much less than c and tends to infinity as v approaches c.

The difference is shown in the attached plot where the non-relativistic definition is the dotted line. There is no need to postulate expansion of space in order to explain redshifts greater than one, in fact you get z=1 for a velocity of just under 180,000 km/s which is quite modest and probably will be achieved by human artefacts sometime in the next few thousand years.

This isn’t just of academic interest. In my day job I’m involved in tracking spacecraft which have quite high radial velocities relative to the Earth. Both Solar Orbiter and Bepi-Colombo can exceed 50 km/s at certain points on their orbit. These spacecraft downlink using a radio frequency of 8GHz (8E9 Hz). Using v/c to predict the Doppler in this case gives an error of almost 120 Hz which is huge.

• This topic was modified 2 years, 5 months ago by Nick James.

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[Robin Leadbeater]

Actually the formal definition of redshift z (eg as quoted for a particular object in the literature) does not involve velocity at all. It is determined by direct measurement z = (measured wavelength/rest wavelength)-1. It is when this is interpreted as a velocity without considering relativity, the frame of reference and the particular cosmological model that the problems occur

[Robin Leadbeater]

In times of crisis, I find myself turning to Prof Ned Wright’s tutorials. These FAQ are useful here

What is the redshift ? https://www.astro.ucla.edu/~wright/cosmology_faq.html#z
Can objects move away from us faster than the speed of light? https://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL
Are galaxies really moving away from us or is space just expanding? https://www.astro.ucla.edu/~wright/cosmology_faq.html#MX

[Nick James]

That’s true when z is cosmological but the point I was trying to make is that you can have z > 1 without having to invoke the expansion of space or any non-flat geometries. The article implies that you can’t.

[Dr Andrew Smith]

I think that Robin is right in advocating the operational definition of red shift as fundamental.

In reality this is what we measure and the interpretations are model dependant (cosmological, relative velocity or gravitational for example). They are often mixed together and only individually present in simple situations.
Regards Andrew

[Dr Paul Leyland]

Nick James wrote:

There is no need to postulate expansion of space in order to explain redshifts greater than one, in fact you get z=1 for a velocity of just under 180,000 km/s which is quite modest and probably will be achieved by human artefacts[sic] sometime in the next few thousand years.

I am not sure whether you consider tightly collimated clumps of electrons, positrons and protons as “human artifacts”. I do, because they do not occur naturally as far as I know, especially those of the positron variety.

If you do include such things, we achieved such velocities many decades ago.

[Nick James]

That’s true but I was thinking of something with a bit more mass and capability like a small interstellar space probe. Accelerating charged particles to a significant proportion of the speed of light is relatively easy. Accelerating a spacecraft, even a very small one, is a lot harder.

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