exupero's blog

View from a photon

It's said Einstein came up with the theory of relativity by imagining himself riding alongside a beam of light. Given what we now know about relativistic physics, it's hard for me to imagine Einstein would have seen very much at all.

In particular, I tend to think in terms of the limit of Lorentz contraction. As a traveler moves at relativistic velocity, they see stationary objects contract lengthwise. This is explored in Poul Anderson's novel Tau Zero, in which a runaway spaceship accelerates faster and faster, eventually reaching such relativistic velocities that it passes through entire galaxies in fractions of a second: because of the spaceship's tremendous speed, a galaxy's thousands of light-years is contracted to mere meters.

The relationship between velocity and relativistic contraction is modeled by this Lorentz equation:

in which L is the observed length, L₀ is the non-contracted length, v is the observer's velocity, and c is the speed of light. When the observer's velocity is equal to c, the observed lengths of stationary objects is zero. Thus, while Einstein may have imagined riding alongside a beam of light and making observations of it, the lengths of everything else in the universe would be contracted to zero length. That is, the universe would appear two-dimensional.

To me, then, this seems to a photon's view of the universe. Everything from its point of origin to its final destination is squeezed together into two dimensions. It doesn't see itself as traveling through space but as existing at a fixed point in a plane, surrounded by flattened versions of whatever it interacts with. It may abut several mirrors, but the concept of bouncing off them is meaningless. A photon would seem to exist in a surreal, oddly distorted version of Flatland.

Of course it's unreasonable to anthropomorphize a photon with sensory input, and relativity is counterintuitive enough that it's difficult to know whether such a thought experiment has any rigor. Beyond those obvious challenges is the complication that the equation for Lorentz contraction has removed a reciprocal. The equation above is derived from applying the Lorentz factor, γ(v), to length, specifically dividing length, then simplifying:

But the Lorentz factor is undefined when v equals c due to dividing by zero. The Wikipedia page on Lorentz transformation states, "The value of v must be smaller than c for the transformation to make sense." But no such division occurs in the formulation of Lorentz contraction. So is Lorentz contraction at the speed of light undefined due to an undefined Lorentz factor, or is contraction infinite because division by zero can be written out of the equation?

I'm not enough of a physicist to have figured this out yet, so if you've wondered about something similar or you can clarify my amateur's understanding, please email me.