# 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?

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!

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.

• afaik there is no experimental evidence that the photon is the source of a gravitational field. – Helder Velez Mar 30 '15 at 12:08
• @HelderVelez due to the stress-energy tensor, photons have a gravitational field (albeit small) – Jimmy360 Mar 30 '15 at 12:09
• If the electron and the photon are created at the same time then the photon will never acknowledged the gravitational field because this is expanding also at c speed. The Stress Energy tensor will change as time goes by? or when the photon enters or leaves a growing grav field region? OR: What happens to the electron and to the photon energy as their respective gravitational field (i.e. energy) expand away from them ? My answer is that they will have to loose energy to source their fields. – Helder Velez Mar 30 '15 at 12:42
• Since the photons energy goes down so will it's gravitational field strength as time goes by. – Jimmy360 Mar 30 '15 at 12:46
• @HelderVelez so as time goes the gravitational force gets – Jimmy360 Mar 30 '15 at 12:48

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.

• it would be red-shifted, since it is moving away from the electron – Jimmy360 Mar 30 '15 at 11:54
• @Jimmy360: correct, tank you. – Helder Velez Mar 30 '15 at 12:01
• Ummm..... Electrons have mass and energy. Therefore, they do source a gravitational field. – Jim Mar 30 '15 at 15:33
• @Jimnosperm : see Jerry's answer to PSE-here. In short: we have an expectation based on theory but we have not 'experimental evidence'. – Helder Velez Mar 30 '15 at 16:33
• You're right, we have yet to devise a sensitive enough experiment that could probe the gravitational field generated by an electron. But the probability of it not generating any gravitational field is so low that mentioning it at all borders on being intentionally misleading. – Jim Mar 30 '15 at 16:41

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.

• Sorry to nitpick, but see this Baez article and note this: Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. Also see Einstein's Leyden Address where you can read this: "empty space" in its physical relation is neither homogeneous nor isotropic... Space isn't curved where a gravitational field is. It's inhomogeneous. – John Duffield Mar 30 '15 at 12:10
• When I said "space", I mean of course spacetime which is why I am talking about "geodesic distance" and not spatial distance. But you are right to point out that the formulation was not very clear, I will clarify. You can see how the shift in the geodesic distance and gravitational redshift are connected in this note. – physicus Mar 30 '15 at 17:47
• I took a look. With respect, the gravitational redshift section is arguably misleading. Photons appear to get redder if they go up, and appear to get bluer if they go down. But conservation of energy applies. There is no mechanism by which the photon gains or loses energy. The section on the Shapiro delay is debateable too. Take a look at the original paper and you can read this: "Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path". – John Duffield Mar 30 '15 at 18:51
• What do you mean with "appear to get redder", this is what actually happens. And what do you mean with the energy being conserved? For energy conservation we need time translation invariance of the background. If we neglect the backreaction of the photon stress-energy on the gravitational field, time translation invariance is indeed obtained, but then the energy the photon loses during redshift due to the dispersion relation, while moving away from the electron, is just transformed into potential energy in the effective gravitational potential due to the electron. – physicus Mar 30 '15 at 19:30
• You should be careful with arguments involving energy conservation in GR, see e.g. this physics.SE post – physicus Mar 30 '15 at 19:32

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.