I have an issue with how long sunlight takes to reach Earth

Question

It is a well-known fact that light from the Sun takes about 499 seconds to reach Earth. However, there is such a thing as time dilation; different observers experience time differently due to there existing between them either a relative velocity (or a difference in in gravitational potential) between their locations. In the case of sunlight headed towards Earth, the different observers are the photon from the Sun and the human on Earth, right? So, when we say sunlight takes 499 seconds to reach Earth, that is elapsed time according to which observer? The photon or the human? If it is the photon, then what is the human's elapsed time? If it is the human, then what is the photon's elapsed time? What am I missing?

Answer 1

I believe the time is measured simply as being the distance between the Earth and the Sun (in our reference frame) divided by the speed of light.

In general, an object travelling from the Sun to the Earth would measure some time $\Delta t'$ elapse when $\Delta t$ has elapsed on the Earth, with $\Delta t' = \gamma \Delta t$, the standard time-dilation formula. As $v\to c$, the $\Delta t'$ will be seen to approach (but never reach) zero. If you plug in $v=c$, the mathematics tells you that $\Delta t' = 0$, and you might be tempted to say that "no time" passes for the photon.

However, as JonCuster points out in the comment above, any object moving at $c$ does not have a rest frame according to Special Relativity, as there is no way for one to get to a velocity of $c$ using a finite number of Lorentz Transformations. (See Does the photon have a rest frame? for explanations.) As a result, treating a photon like any other inertial observer is not allowed, and so questions like "What time interval did the photon experience?" or "How far away did the photon perceive these two points to be?" do not make any sense.

Answer 2

If you just consider special relativity, then you are asking about the time difference between 2 events: The emission of a photon at the Sun ($A$), and the arrival or detection of that photon at the Earth ($B$). Since their separation is light-like, you can find coordinate systems in which $(t_B - t_A) \rightarrow 0 $, which is not very useful. On the other hand, you can us 499 seconds, which is maximum time difference as it is measured in the frame in which the distance between the Sun and Earth is maximized (their relative rest frames).

Further inclusion of gravitational effect complicates the matter.

There are many instances where astronomers, spacecraft navigators, and mission planners are concerned about the time difference between two events in the solar system. In the case of a photon traveling from the Sun to the Earth, it's save to just use UTC time differences and ignore relativistic effects, they are small. If your planning an engine burn for an orbit insertion, a coherent radar scan of Titan's surface, or some other activity requiring high precision, you need to consider both special and general relativity.

The current standard is called "Barycentric Dynamical Time" (https://en.wikipedia.org/wiki/Barycentric_Dynamical_Time). The linked article says:

"Barycentric Dynamical Time (TDB, from the French Temps Dynamique Barycentrique) is a relativistic coordinate time scale, intended for astronomical use as a time standard to take account of time dilation when calculating orbits and astronomical ephemerides of planets, asteroids, comets and interplanetary spacecraft in the Solar System. TDB is now (since 2006) defined as a linear scaling of Barycentric Coordinate Time (TCB). A feature that distinguishes TDB from TCB is that TDB, when observed from the Earth's surface, has a difference from Terrestrial Time (TT) that is about as small as can be practically arranged with consistent definition: the differences are mainly periodic,and overall will remain at less than 2 milliseconds for several millennia."

The point here is that the definition of the time difference between 2 events in the solar system is not a trivial matter; rather, it has a long and technical history.

Answer 3

The human and all observers whose relative velocity to Earth is not relativistic ($ v \ll c $). The photon's elapsed time is zero.