# What part of special relativity is factored in in relativistic redshift velocity equations?

While doing research into redshift equations (doppler redshift and cosmological redshift), both types of redshift had two equations for finding recession velocity: a 'non-relativistic' equation and a 'relativistic' equation. I looked up SR, and found out about time dilation and length contraction. My question is which of these phenomena have been factored in in relativistic equations (maybe both, or possibly a third phenomena I'm unaware of?)

I'm doing this for a high school project, so please explain in very simplistic language. Also showing a bunch of complicated equations will seriously fly right over my head, so I'd greatly appreciate it if you could please use words only :)

Speaking of Special Relativity, there are three basic relativistic effects: time dilation, length contraction, and relativity of simultaneity. Of those three the relativity of simultaneity is the most difficult for new students to understand, and it is the source of most of the errors that new students make in relativity.

These three effects are combined in the Lorentz transform. All of the relativistic formulas can be derived from the Lorentz transform. So all the relativistic formulas account for all three effects: time dilation, length contraction, and relativity of simultaneity.

• ".....can be derived from the Lorentz transform". Exactly. We must use the Lorentz transformations to solve our problems like this one. Time dilation, length contraction, and relativity of simultaneity must not be used to solve problems but must be used for the interpretation of the results after applying the Lorentz transformations. – Frobenius Oct 19 '18 at 10:38
• I agree. Personally, I think that teachers should teach the Lorentz transform primarily. Time dilation and length contraction and the relativity of simultaneity naturally come out of the Lorentz transform whenever appropriate, and inappropriate application of the simplified formulas is avoided. – Dale Oct 19 '18 at 10:52
• Could you please explain the effects of time dilation, length contraction and relativity of simultaneity on redshift? I think I may understand how time slowing down would change velocity, since velocity is m/s, but my brain is just really muddled right now. I'm also very confused as to how simultaneity being relative has any impact on redshift? Thank you so much :) – Lolo123 Oct 19 '18 at 12:47

The simplest case is the "ordinary" relativistic Doppler redshift or blueshift due to relative motion.

Considering the relativistic effect (redshift or blueshif), it is necessary to take into account the time dilation. At relativistic velocities all processes in moving source or moving observer run slower, thus a source oscillates slower and this effect must be added to the classical one.

In simple terms: take the equation for the classic Doppler effect and "attach" time dilation:

Due to time dilation, the relativistic effect will be more reddish than the classic one. Hence, if you consider the source to be moving; the equation will look like that:

$$f_r = \frac{f_s}{\gamma\left(1 + \beta \cos\theta_r\right)}$$

Or to the detector, if you consider the detector to be moving in the reference system of the source.

$$f_r = \gamma \left( 1 - \beta \cos \theta_s \right) f_s$$

here $$\gamma = \frac{1}{\sqrt{1 - \beta^2}}$$ is the Lorentz factor.

If source and receiver moving directly towards or away from each other, by simple transformations, these equations can be reduced to a general form

$$\frac{f_s}{f_r} = \sqrt{\frac{1 + \beta}{1 - \beta}}$$

Observed frequency shift is invariant, it doesn't depend on choice of frame,even though relative contribution of time dilation are frame - dependent.

The Feynman Lectures - Relativistic Effects in Radiation

• Hi, thank you for answering :D So in short time dilation has the largest effect on redshift? Would length contraction also impact? – Lolo123 Oct 19 '18 at 12:50
• Ar low ordinary velocities contribution of time dilation is negligible, at high relativistic velocities it is very significant. As nice example, in the frame of "stationary" source, if observer moves towards it with velocity close to c, maximum "classical" frequency he can see would be 2f. However, due to dilation of observer's clock he will "see" this frequency as infinitely high. You can undersrand this problem with time dilation. For some advanced treatment please look for Relativistic Doppler Effect in Wikipedia. – Albert Oct 19 '18 at 13:34