# Acceleration and its effect on the speed of light

One of Einstein's thought experiments involved a rocket and two light sensors inside the rocket, one at the top, and one at the bottom. When the rocket was at rest, a pulse of light from the bottom sensor travels to the top sensor at the speed of light, $c$. But when the rocket is launched and is accelerating, it takes the pulse of light longer, or $c_-$, to travel from the bottom sensor to the top sensor than when the rocket was at rest.

Does this mean that light is slowed down by the increased pull of gravity on the pulse of light?

• Speed of light is affected by gravity, but this observation does not support it. It rather supports speed of light is finite and relative speed of light could be changed. – Anubhav Goel Jan 19 '16 at 2:24
• what do you mean by "or c-" ??? – user46925 Jan 19 '16 at 3:03
• What gravity would the light be exposed to in this Gedankenexperiment? – CuriousOne Jan 19 '16 at 4:07

The local speed of light is always $c$. By this I mean that if any observer anywhere measures the speed of light they will get the value $c$. It doesn't matter if they are accelerating or not.

However for an accelerating observer, even in flat spacetime, the speed of light at distant locations will not be $c$. The geometry of spacetime for an observer with constant accelerating $a$ is described by the Rindler metric:

$$c^2d\tau^2 = \left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 - dx^2$$

We get the speed of light from this buy setting $d\tau = 0$, which gives us:

$$\frac{dx}{dt} = c\,\left(1 + \frac{a}{c^2}x \right) \tag{1}$$

For negative $x$, i.e. in the opposite direction to the acceleration, the right hand side of equation (1) is less than $c$ so in this direction the velocity of light decreases. In the other direction the velocity of light increases.

This is essentially the same phenomenon described in Does gravity slow the speed that light travels?. For a stationary observer in a curved spacetime the geometry looks locally like Rindler space. Note the caveats I make in the linked question. The quantity calculated here is the coordinate velocity of light not the velocity measured by any observer.

• Too much math for me,,but does it make sense that if lght can be and is bent by the sun's gravity when it is passing side-to side to the sun, light can be retarded by a gravitational force acting vertically downward on light moving vertically upward? – Roger Jan 20 '16 at 2:47
• @Roger: I'm afraid your intuition is leading you astray. For example you might think that light falling towards a black hole would speed up, but in fact it slows down. – John Rennie Jan 20 '16 at 6:06
• What is meant by "the speed of light at distant locations"? – aquirdturtle Jan 20 '16 at 8:09
• @aquirdturtle: suppose you set up some coordinates $(x, y, z)$ with yourself at the origin i.e. in these coordinates you are positioned at $(0,0,0)$. The speed of light at your position, $(0,0,0)$, is always $c$. However at any other position in your coordinate system $(x\ne0,y\ne0,z\ne0)$ the speed of light may not be $c$. A distant location simply means any position $(x\ne0,y\ne0,z\ne0)$. – John Rennie Jan 20 '16 at 8:33
• I think you misunderstand what I was getting at. I'm used to thinking about local measurements of the speed of light, e.g. measured by clocks being placed everywhere which each measure local time. How do you measure the speed of light at a distant place in a way that's different than this? – aquirdturtle Jan 20 '16 at 8:36

If we ask a bottom observer, he says that when the gravity field gets stronger, it takes the pulse of light a shorter time to travel from the bottom sensor to the top sensor than when the gravity field was weaker, because of the gravitational time dilation that slows down his clock and slows down the light, but slows down more his clock than the light.

If we ask a top observer, he says that when the gravity field gets stronger, it takes the pulse of light longer time to travel from the bottom sensor to the top sensor than when the gravity field was weaker, because of the gravitational time dilation that slows down his clock and slows down the light, but slows down more the light than his clock.

If we ask a launchpad observer, he says that when the rocket starts to accelerate, it takes the pulse of light longer time to travel from the bottom sensor to the top sensor than when the the rocket was at rest, because the sensor is trying to flee the light pulse, so to speak.

Paradoxically enough, the speed of light in a vaccum is completely unchanged by any outside factors. It doesn't matter what speed the object is moving, the speed of light is always the same in any frames of reference.

• What do you think this adds over the answers already present? – ACuriousMind Jan 19 '16 at 23:32
• At least in my humble opinion, it presents the answer in simpler terms more accessible to the physics-illiterate. Just a thought, feel free to disagree. – DevilApple227 Jan 19 '16 at 23:36

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