I know light's speed in vacuum is constant, but what about its velocity?

Question

Is the velocity of light in vacuum constant? It seems it would be different depending on whether it is coming toward you or away from you, but I just want to make sure. Does the direction of light change from one frame to another frame?

Answer 1

Light can obviously travel in any direction, but the magnitude of its velocity (in vacuum) is always $c$.

The magnitude of the velocity is a scalar i.e. just a number, but the velocity is a vector. To specify the velocity we need to choose some axes. For example I might choose the Cartesian axes $x$, $y$ and $z$. In that case light approaching me from the positive $x$ direction would have the velocity $(-c, 0, 0)$, while light moving away from me in the positive $x$ would have the velocity $(c, 0, 0)$. These velocities are different vectors, but they both have the same magnitude of $c$.

To be more precise the local velocity of light, i.e. the velocity you measure at your location, always has a magnitude of $c$. The magnitude of the velocity at locations distant from you can be greater or less than $c$ even in special relativity.

Answer 2

In a very real sense, the velocity of a light ray in a curved spacetime is constant, or at least as constant as it can be; this is because it follows a special path in spacetime called a geodesic.

The problem with defining a "constant" vector on a curved surface (the surface of the Earth, say) is that you can't easily compare tangent vectors at two distinct points on the surface; roughly speaking, the tangent planes point in different directions.¹ We get around this by defining a notion of "parallel transport". Given a vector at a particular point on a curved manifold, we move it an infinitesimal step along a particular path. Since this step is infinitesimal, there can be a well-defined notion of how the two tangent planes at these two infinitesimally separated point are related to each other. We can then take another infinitesimal step along that path, update our tangent vector, and repeat (i.e. integrate) until we have mapped the vector at the original point to another vector at another point on the manifold. Along this path, the vector is "as constant as possible", since we're changing the vector as little as possible at each infinitesimal step. We have to change the vector some—it's a curved manifold—but there's a well-defined sense in which you can say that two vectors living at infinitesimally close points are basically equal to each other.

A geodesic is then a path whose tangent vector is "as constant as possible", in the following sense: given a starting point and a vector at that point, take an infinitesimal step in the direction of the vector, and parallel-transport the vector to that new point. Then take another infinitesimal step in the direction of the vector, and parallel-transport that vector. Since the tangent vector of the curve is parallel-transported along that curve, then we can consider the tangent vector to be "as constant as possible" along the path.

So any particle that follows a geodesic in a curved spacetime is, in a very real sense, moving with "constant velocity" through spacetime (or at least as constant a velocity as the spacetime allows). Light rays move along geodesics, but as it happens so do massive particles (without other forces acting on them) move along geodesics too.

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¹ This isn't the best way of describing this notion, since it depends on the embedding of the sphere in a higher-dimensional space. Rest assured that it can be defined in a more mathematically rigorous way as well.

Answer 3

Yes light does have different directions in different frames. Two observers with different velocities will see the same photon traveling in different directions.

One observer standing still at noon sees light traveling vertically downward. Light that strikes the top of his head would also strike his toes.

An observer running forward see light slanted backward. Light that strikes the top of his head takes time to reach his toes. By that time, his toes have moved forward. The light lands a short distance behind him. Thus the angle is $\sin\theta = \dfrac{v}{c}.$

Answer 4

I know light's speed in vacuum is constant, but what about its velocity?

The speed of light in vacuum is not constant, and because of this light curves, hence its vector-quantity velocity varies. Have a look at the Einstein digital papers and you can find Einstein talking about it:

This is what John Rennie was referring to. Think about the room you're in, and appreciate this: light goes slower near the floor than at the ceiling. If it didn't, optical clocks wouldn't go slower when they're lower, light wouldn't curve, and your pencil wouldn't fall down.

However for some strange reason most people don't seem to know about this, even though there's plenty of places you can read about it. See for example Irwin Shapiro's 4th test of General Relativity, along with The Deflection and Delay of Light by Ned Wright, or Is The Speed of Light Everywhere the Same? by PhysicsFAQ editor Don Koks. There seems to be some kind of issue with current teaching wherein the speed of light is usually taken to mean the locally measured speed of light rather than the "coordinate" speed of light. The locally-measured speed of light is always the same because of a tautology, see http://arxiv.org/abs/0705.4507. We use the local motion of light to define the second and the metre, which we then use to measure the local motion of light. So we are guaranteed to always measure the same value.

Answer 5

I know light's speed in vacuum is constant

Correct. Specificly, we speak of "vacuum" (and evaluate "refractive index" value $n = 1$) in the context of signal exchange if phase speed and group speed of the "signal carrier" are equal to the signal front speed $c_0$;
where $c_0$ is just a particular (non-zero) symbol which appears in the chrono-geometric definition of distance (between "ends" which remained at rest wrt. each other),
and where "signal front" is presumably an unambiguous notion.

Is the velocity of light in vacuum constant?

Comparison in terms of velocity involves of course comparison in terms of speed, and comparison in terms of direction. Since signal front speed is unchangeable by definition, let's consider possible variability in direction. There are two aspects, related to what we mean by "direction" in the first place:

coming toward you or away from you

This illustrates that "direction" of signal exchange is foremost described by the (distinguishable) identities of signal source (e.g. "you", and/or "$A$") and signal receiver (e.g. "$B$"); and that

• "$B$ having observed a signal indication of $A$"

describes signal exchange in the opposite direction than

• "$A$ having observed a signal indication of $B$";

"$A$ to $B$" vs. "$B$ to $A$". In general, therefore, it is said that different signal exchanges may "have proceeded in different directions".

Another consideration has to do with the notions of "straightness" and "line of sight":
If, for instance, participant $A$ had stated a partcular signal indication, $A_*$, and participant $B$ observed this signal (indication $B_{\circledR A*}$), and a further participant, $F$, observed $A$'s signal indication in coincidence with having observed $B$'s indication of having observed $A$'s signal indication (indication $F_{\circledR A*} \equiv F_{\circledR B \circledR A*}$)
then these three signal exchanges ("$A$ to $B$", "$A$ to $F$", and "$B$ to $F$") are said to have been in the same direction.

In this sense, any one specific completed signal exchange can be said to have "proceeded throughout in the same direction (from signal source towards receiver)", and therefore at constant velocity; "along a light-like geodesic".

Does the direction of light change from one frame to another frame?

The direction of a specific completed signal exchange is defined by the (distinguishable) identities of signal source and signal receiver; which remain independent of any particular choice of reference system (with respect to which the trajectories of source and receiver may be described).

Answer 6

Constancy of the speed of light

When one asks about the speed of an object, it is important to first define in what frame of reference you are measuring that speed. If Car A and Car B are both speeding at 100mph and heading directly towards each other, the police officer sitting at the side of the road may measure each of them at 100mph in his frame, but the driver in Car A will see Car B heading towards him at 100mph +100mph=200mph. So clearly the frame of reference in which we measure something, makes a big difference to a measured speed. However, interestingly, this is NOT true for light (let's assume its traveling in a vacuum). If my car is stationary on the road and I measure the speed of a laser beam (with a device in my car) and a bullet reaching me from my enemies car (also with a device in my car), and my enemies car is also sitting stationary further up the road, I will measure the laser beam to be traveling at c towards me, and the bullet at (say) 500mph towards me. If I start traveling towards my enemy at 100mph, I will STILL measure the laser beam to be traveling at c, but the next bullet that he fires when I am moving will appear to be traveling at 500+100 mph =600mph towards me.

So, at least in a vacuum, no matter the relative speed of the observer and the emitter, the light will always be measured to travel at c in EVERY frame of reference.

People may then ask why light appears to travel slower than c when passing through a transparent medium like glass and why its time to pass through the medium appears to be affected by my speed relative to the medium. In fact, all measurements show that light continues to travel at c through the medium. However, when passing through the medium, it is absorbed by the molecules in the medium and then re-emitted, and it is this absorption and emission process that adds time to the total transit time through the medium. But when the light is re-emitted and traveling to the next molecule, it again travels at c between the interactions with the molecules. So the physical speed of the light is not really reduced in a medium even if the transit time of the signal varies from one material to another depending on the refractive index. It is the net absorption to re-emission time that varies. Once the photon comes back into existence, even between absorption events, its speed will appear to be c. Also, relativity will appear to modify the apparent thickness of the glass if I am moving at a high speed relative to the glass. All these effects have to be taken into account, but interestingly, no matter how much these change due to relativity, the final calculated speed of light in a vacuum is c.

Light also appears to travel slower than c in some circumstances when observed in space, particularly near a black hole event horizon. But the reason for this is not strictly that the light is slowing. The clocks are slowing down, and the space is becoming infinitely curved. The curvature adds an extra distance to the trajectory of the light that we can't easily see, so that it appears to be traveling slower. But if we assume it is NOT traveling slower, that is what allows us to calculate the extra distance from the curvature! Also, in the presence of intense gravity near the event horizon of a black hole, theory tells us that our clocks will appear to slow down due to gravity. But distance is also changing, so, interestingly, the calculated speed of light remains constant. This is what actually creates the event horizon. Time has slowed down to near zero, but the distance has also dramatically shrunk and curved in on itself, so, to all intents and purposes, light continues to travel at the same speed, although many of its other parameters are dramatically altered.

We also have to be very careful with these calculations in circumstances where relativistic effects kick in. If we are traveling at a very high speed w.r.t. the glass or other medium, relativity can make its apparent thickness change and change how fast the clocks being used for time measurement are running. If we don't take these relativistic changes into account, then it will make the speed of light appear to be changing. Of course, Einstein calculated all the relativistic equations by ASSUMING that the speed of light remains constant, so was forced to account for changes in the transit time by relativistic changes in distance and time. However, his assumption has been proven, experimentally, to be correct.

It was, in fact, this assumption by Einstein that explained why it was not possible to travel faster than the speed of light. As we start to move faster in some frame of reference, we find that the time starts to slow down in that frame of reference (compared to our moving frame), and the physical size of the other frames starts to change. This may not be intuitively obvious, but relativity allows us to combine both these effects and to calculate them. What Einstein's calculations showed was that, as we travel closer and closer to the speed of light, the clocks that we use to measure the time gradually slow down, and if we actually reach the speed of light, the clocks would slow to zero, so we would appear to be there for eternity according to those clocks. This created an "infinite" time barrier that would prevent us from traveling at the speed of light relative to another frame of reference.

One point I should mention is that this does not actually prevent us from traveling faster than the speed of light. It simply prevents us from initially traveling slower than the speed of light and then accelerating past the speed of light. If some physical effect somehow created a particle that came into existence traveling faster than c, that particle would see the barrier from the other side and would not be able to slow down to drop below the speed of light. These are called "Tachyons" in the world of particle physics. Not a lot of physical evidence to support their existence at this point, but the theory exists.