In $1$-dimensional space, how would the gravity generated by an electron affect a photon moving away from the electron if the photon can’t slow down?

Question

Suppose we had a universe obeying the same physical laws as our own. But it had only one spatial dimension (represented by the $x$ axis) and it was totally empty. There are just two point-like particles in this universe:

• An electron which is at rest.
• A photon which is moving away from the electron.

Yet we have two important rules that can’t be broken:

• A photon can’t slow down, its speed must always be equal to $c$.
• Gravity affects all form of matter, even photons.

.

So how would the gravity engendered by the electron affect the photon if it can’t slow down?

If this was in $3$-dimensional or $2$-dimensional space, there would be no problem since the photon could just be slightly deviated from its trajectory. But here the photon is moving away from the electron very precisely along the axis joining them, we’re in $1$-dimensional space, the photon can’t be deviated.

We've got a paradox over here!

Answer 1

The energy of a photon is given by the equation E = hf where h is Planck's constant and f is frequency. The energy would decrease, making the frequency decrease (since h is constant). So, if the photon was blue light, then it would get redder and redder as time when on. There is a point, however, when your system eventually stops working. This is because the photon actually exerts a gravitational pull on the electron so eventually it would start moving. This doesn't change the answer, but it means your system cannot be maintained as stated, indefinitely. The electron will start moving . Photons exert a gravitational pull bacease of their contribution to the Stress Energy Tensor.

Answer 2

The energy (i.e. frequency) of the photon will change as it travels thru a gravitational gradient, by Einstein and so: as the photon goes away its color will be redshifted.

What experiment proved that the electron is the source of a gravitational field? none afaik.

edit post:
the total energy budget ( electron + grav field + photon) is compromised if the answer is restricted to the above sentence.

Answer 3

Two thoughts:

1. You have to be careful about what you mean with the "same physical laws as our own" in one spatial dimension. If you write down Einstein gravity in 1+1 dimension you realize that it is completely topological, i.e. there are no local excitations, no gravitons, no equations of motion etc. It is not surprising then that this theory looks completely different than Einstein gravity in 3+1 dimensions.

2. Even though the photon cannot change its direction, it can still of course be influenced by the spacetime curvature due to the presence of the electron. For example, it could take the photon longer to cover a certain distance if spacetime is curved even though the speed is of course still $c$. The geodesic distance between two points just becomes altered due to the gravitational field. The change in the geodesic distance, together with the null condition, i.e. the condition that the photon travels at the speed of light, results in a gravitational redshift, which can be worked out using the geodesic equation as laid out in this lecture note.

Answer 4

So how would the gravity engendered by the electron affect the photon if it can’t slow down?

Somewhat counter-intuitively, the ascending photon speeds up in line with the increasing "coordinate speed of light". See this PhysicsFAQ article where Don Koks writes this:

"Given this situation, in the presence of more complicated frames and/or gravity, relativity generally relinquishes the whole concept of a distant object having a well-defined speed. As a result, it's often said in relativity that light always has speed c, because only when light is right next to an observer can he measure its speed— which will then be c. When light is far away, its speed becomes ill-defined. But it's not a great idea to say that in this situation "light everywhere has speed c", because that phrase can give the impression that we can always make measurements of distant speeds, with those measurements yielding a value of c. But no, we generally can't make those measurements. And the stronger gravity is, the more ill-defined a continuum of observers becomes, and so the more ill-defined it becomes to have any good definition of speed. Still, we can say that light in the presence of gravity does have a position-dependent "pseudo speed". In that sense, we could say that the "ceiling" speed of light in the presence of gravity is higher than the "floor" speed of light."

Note that whilst we talk about gravitational redshift and blueshift, the photon doesn't actually lose or gain any energy. You can appreciate this if you imagine sending a 511keV photon into a black hole. The black hole mass increases by 511keV/c². You measure a photon as blueshifted when you descend because you lose energy.