Everting the Doppler Shift

An astronomer has determined the red shift of a distant galaxy by measuring the wavelength of one of the bright lines in the spectrum of light emitted by the galaxy's hydrogen content and dividing it by the wavelength of the corresponding line measured in a laboratory on Earth to obtain the ratio γ. Because that galaxy moves almost directly away from Earth, the astronomer knows that the simplest form of the relativistic Doppler shift accounts for the red shift, so they have

(Eq'n 1)

Now our astronomer wants to calculate the speed v at which the galaxy flies away from Earth:

1. square the equation;

(Eq'n 2)

2. multiply and divide by c^{2};

(Eq'n 3)

3. factor the denominator and cancel the common factor of c+v in both the numerator and the denominator;

(Eq'n 4)

4. multiply by c-v;

(Eq'n 5)

5. add
γ^{2}v and subtract c;

(Eq'n 6)

6. divide by 1+γ^{2};

(Eq'n 7)

Q.E.I.

Now our astronomer can substitute that formula into the formula for the Lorentz factor to calculate the amount of relativistic distortion in readings from the galaxy,

(Eq'n 8)

I leave the working out of the details of that calculation to the reader.

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