# If curved paths imply that the vehicle is accelerated, how come do we assume that light gets curved whilst its speed is constant?

I don't understand how we can accept these two sentences at the same time:

1. Light speed is constant, therefore experiences no acceleration.

2. On the presence of a gravitation field, light path is curved (which would mean that somehow it gets accelerated).

• From the photon's perspective, it is going in a straight line and not being accelerated. General relativity is not a simple Euclidean space. – Jon Custer Nov 20 '15 at 22:18
• Ok, good point. Could you please explain further? Many thanks. – Iago Rodríguez-Quintana Nov 20 '15 at 22:57
• @JonCuster That is not the point: the answer is that, although the speed must be constant, the direction of the electromagnetic waves (light) may change. – gented Nov 20 '15 at 23:40
• I am sorry.. So finally you explain it by changing the direction, although keeping speed constant.. That is mainly what the others have said, and I am not able to understand that concept. Light travels isotropically and constantly, so I resist to accept direction changes... – Iago Rodríguez-Quintana Nov 21 '15 at 0:21
• Because direction changes would imply acceleration somehow. – Iago Rodríguez-Quintana Nov 21 '15 at 0:23

## 2 Answers

What you are confusing here is speed and velocity. Light speed is constant, but the velocity, which takes into account the direction as well as the speed is not. As an example of how something can accelerate without changing speed, consider the case of circular motion, where the acceleration of an object moving at a speed $v$ in a circle of radius $r$ is $a=v^2/r$. In this case, the speed is constant, but the velocity is changing constantly. Gravity can also cause light to change its direction, and therefore its velocity. But the speed of light will remain constant in all reference frames. Its the law :)

• I understand that what you say is that velocity would be a vector (which considers direction), the magnitude of which is speed. So if we split the vector in its dimensions, there would appear some shorter vectors on each direction: meaning that each of them wouldn't respect speed of light, which should be constant, regardless the direction. Thanks but I still don't get it. – Iago Rodríguez-Quintana Nov 20 '15 at 21:44
• @IagoRodríguez-Quintana Right, it should be constant regardless of direction, but each component of the light velocity doesn't need to be equal to the speed of light. Instead, the vector sum of all of the components needs to be equal to the speed of light, like $c = \sqrt{v_x^2 + v_y^2 + v_z^2}$. There is no requirement that $c = v_x$, $c = v_y$ or $c = v_z$. – tmwilson26 Nov 20 '15 at 21:46
• Really? It doesn't travel constantly and at that magnitude on every direction? I am kind of surprised! Could you explain yourself further? – Iago Rodríguez-Quintana Nov 20 '15 at 21:58
• Light is directional, in that it carries momentum in the direction that it travels. The light from a laser aimed at a wall travels with a velocity component towards the wall and not in directions perpendicular to that, where the velocity component in those directions will be zero. A gravitational field can change the path of light, moving velocity from one component to the other, but the vector sum of all of the components remains fixed. – tmwilson26 Nov 20 '15 at 22:06
• You should think about the individual photons velocity as opposed to the direction of all of the light from the source. Each photon travels in a particular direction, but there are many photons travelling in many directions – tmwilson26 Nov 20 '15 at 22:33

The disconnect is between the first and second clauses of your first sentence:

Light speed is constant, therefore experiences no acceleration

Yes, the speed of light is a constant, but it experiences no acceleration in its direction of travel. Light definitely accelerates laterally when gravity pulls on it, which is why it curves when passing near massive objects.

• Thanks for the answer. But I still don't understand precisely: you state that it travels constantly (in some direction) but also accelerates laterally... So does it accelerate finally, overpassing speed of ligth "laterally"? – Iago Rodríguez-Quintana Nov 20 '15 at 21:29
• I am afraid I still don't get it: if the path it covers is curved, it is longer than the straight line, so it should travel faster to appear to bend.. Thanks anyway... – Iago Rodríguez-Quintana Nov 20 '15 at 21:46
• A curved line is longer than what straight line? Think of a (frictionless) car in neutral, rolling through a parking lot. You turn the steering wheel; does the car speed up or slow down? – Daniel Griscom Nov 20 '15 at 22:19
• I think it speeds up somehow, because you experience some acceleration outwards.. Anyway, a curve is always a longer path given a constant speed. Thanks anway. – Iago Rodríguez-Quintana Nov 20 '15 at 22:23
• If it speeds up, then it's a perpetual motion system. Just go around and around the parking lot, dragging some sort of generator system behind you. Infinite power. (Spoiler alert: not!) – Daniel Griscom Nov 20 '15 at 23:07