10. Light and spacetime

Whose clock and whose distance? #

Proper time is the time measured by a clock along its own timelike worldline. Coordinate time belongs to a chosen frame. The sign convention here is (+,−,−,−). A null interval has zero proper time, but no inertial observer is at rest with a photon. The Doppler formula below describes radial recession in flat spacetime; cosmological redshift uses the scale factor instead.

10.1 The invariant interval #

In flat spacetime,

ds^2=c^2dt^2-dx^2-dy^2-dz^2.

For a massive particle following a timelike trajectory,

c^2d\tau^2=ds^2>0,

where $d\tau$ is proper time. For light,

ds^2=0.

Thus an ideal null worldline has zero proper-time interval.

10.2 Why “from the photon’s point of view” is dangerous #

The Lorentz factor

\gamma=\frac{1}{\sqrt{1-v^2/c^2}}

diverges as $v\rightarrow c$ . No inertial Lorentz frame exists in which a photon is at rest. It is therefore not physically legitimate to transform into “the photon’s reference frame” and ask what the photon sees. Zur Elektrodynamik bewegter K"orper (1905)

A scientifically correct but still poetic sentence is:

Light follows null trajectories for which the spacetime proper-time interval vanishes, but relativity does not define a photon rest frame.

That is the rigorous content behind the popular phrase “light does not experience time.”

10.3 Frequency is observer dependent #

Although $c$ is invariant in vacuum, photon frequency is not. For a source moving directly away at speed $v=\beta c$ , the relativistic Doppler factor is

\frac{\nu_{\mathrm{obs}}}{\nu_{\mathrm{em}}} =\sqrt{\frac{1-\beta}{1+\beta}}.

Cosmological redshift is different in origin and is commonly written

1+z=\frac{a(t_0)}{a(t_{\mathrm{em}})},

where $a(t)$ is the cosmological scale factor.

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