# How far has a 13.7 billion year old photon travelled

I've read that the size of the observable Universe is thought to be around ~46 billion light years, and that the light we see from the most distant galaxies were emitted ~13.7 billion years ago as a result of the expansion. So is the actual distance the photons have travelled 46 billion light years, or 13.7 billion light years?

Age of universe is roughly 13.8 billion years, so anything with longest traveling record can only travel 13.8 billion light years at most (in any direction; not necessarily to Earth) because nothing can travel faster than light.

Now, we are at the center of the observable universe (which is spherical with visible radius 13.8 billion years), so longest traveling record holders are the oldest ones and are at the edge of the observable universe and they have traveled 13.8 billion light years so far.

The actual calculated radius 46 billion light years is different from the visible radius because the universe has expanded.

• They say imitation is the greatest form of flattery. – Rob Jeffries Nov 30 '14 at 14:18

Light always travels at the speed of light when in a vacuum. Space is a pretty good vacuum. So if it's been travelling for 13.7 billion years, then it has travelled 13.7 billion light years.

There is no contradiction here. Yes, those galaxies are now 46 billion light years light years away, but this is because the universe has expanded. You can find lots of questions related to this on Physics SE.

Why is the observable universe so big?

Are the distances we measure to objects in the universe incorrect because of their relative motion?

• So does this mean that the proper distance along the geodesic is 13.7 billion light years while the coordinate distance would be 46 billion light-years, or is it much more complicated than that? – Chris L. Nov 21 '14 at 20:49
• @PetTaxi I might let a cosmologist answer that definitively. I don't think you call the 13.7 billion light years the "proper distance", it is the light travel distance. The proper distance would be 46 billion years now, but smaller in the past. The 46 billion light years is also the (model dependent) co-moving distance, which is constant by definition. – Rob Jeffries Nov 21 '14 at 20:57
• @PetTaxi: The proper length is zero, because the photon's path is lightlike. If you want to define the photon's "odometer," you can't do it in the photon's frame, because a photon doesn't have a rest frame. The 13.7 byr is measured on a clock that's at rest relative to the Hubble flow. The 13.7 blyr is measured on a chain of rulers, each at rest relative to the Hubble flow. – Ben Crowell Nov 21 '14 at 21:16
• @BenCrowell I made your comment a community wiki answer, because it pithily summarises what the OP is after and the concepts they need to research to get a good grip on an answer. – WetSavannaAnimal Nov 23 '14 at 3:23
• @BenCrowell If you do want to record your comment as your own answer, I'll delete mine. – WetSavannaAnimal Nov 23 '14 at 3:24

A community Wiki answer recording someone else's (Ben Crowell's) comment that I think is worthy of being a permanent answer.

The proper length is zero, because the photon's path is lightlike. If you want to define the photon's "odometer," you can't do it in the photon's frame, because a photon doesn't have a rest frame. The 13.7Gyr is measured on a clock that's at rest relative to the Hubble flow. The 13.7Glyr is measured on a chain of rulers, each at rest relative to the Hubble flow.

Here the "Hubble flow" is simply the motion of astronomical objects that is owing wholly to spacetime expansion (defined by the derivative $\dot{a}(t)$ of the scale factor $a(t)$ in the FLRW metric)

Simple calculation: Age of observable universe is 13.8 billion years. Distance light travels in one year is 6 trillion miles. So 13.8 billion x 6 trillion = 8.28e+22. (Mic drop...)