Maxwell's Rainbow — Question 10

Question 10

A supplied redshift law stretches wavelengths and pulse durations together

A propagation model stretches all temporal intervals by a factor $1+z$ while preserving the number of carrier cycles. A source emits a line at vacuum wavelength 480 nm; the observed wavelength is 720 nm. Locally at both endpoints use $f\lambda=c$.

1. Find z and the observed-to-emitted frequency ratio.

2. For an emitted pulse lasting 2.0 ns, find the observed duration and verify conservation of carrier-cycle count.

Question 10 - Solution

A supplied redshift law stretches wavelengths and pulse durations together

(a) Method and calculation

$$1+z=\lambda_{\mathrm{obs}}/\lambda_{\mathrm{em}}=720/480=1.5\Rightarrow\boxed{z=0.50}.$$$$\boxed{f_{\mathrm{obs}}/f_{\mathrm{em}}=1/(1+z)=2/3}.$$

(b) Interpretation and further calculation

$$\boxed{\tau_{\mathrm{obs}}=(1+z)\tau_{\mathrm{em}}=3.0\,\mathrm{ns}}.$$$$f_{\mathrm{obs}}\tau_{\mathrm{obs}}=\frac{f_{\mathrm{em}}}{1+z}(1+z)\tau_{\mathrm{em}}\Rightarrow\boxed{N_{\mathrm{obs}}=N_{\mathrm{em}}}.$$

Physical check

This question applies a supplied stretching model; it does not infer a source velocity from the redshift.