# I Light pulse path, length and shape, when bouncing between two mirrors

#### fog37

Summary
The path of a light pulse moving bouncing between two mirrors (top and bottom) from two different inertial frames.
Hello,
A light pulse moving bouncing between two mirrors (top and bottom) follows a vertical straight path w.r.t. to an inertial observer at rest relative to the mirrors. However, a moving inertial observer see the light pulse move in a zig-zag path as it bounced back and forth between the mirrors...

In special relativity, this zig-zag path produces a larger path length, and, given the constancy of the speed of light, time dilation occurs.

Question: is the path followed by the light pulse straight or zig-zag? Which one is physically true? Both, even if they are different in length and shape?

Thanks

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#### Dale

Mentor
Summary: The path of a light pulse moving bouncing between two mirrors (top and bottom) from two different inertial frames.

Which one is physically true? Both, even if they are different in length and shape?
Yes, both. That is the principle of relativity.

#### Ibix

Question: is the path followed by the light pulse straight or zig-zag? Which one is physically true? Both, even if they are different in length and shape?
I'd say that the nearest thing to a "true" path is the four-dimensional worldline that includes the pulse's travel in the timelike dimension as well. You can project that path on to different three dimensional surfaces to get different "paths through space", but these are simply different descriptions of the worldline.

• Pencilvester

#### PeterDonis

Mentor
Both, even if they are different in length and shape?
They appear different because, as @Ibix said, they are projections into two different spacelike surfaces of the invariant worldline of the light ray through spacetime. Relativity says that "space" is not an invariant, so neither are spatial lengths or shapes of spatial paths.

#### fog37

Cool, thanks, processing it.

A clock is any device that operationally measures time by counting the number of periodic events having a specific duration. In the case of time dilation, the measurement of a time interval gives a larger time if the clock that is being read is inside a moving inertial reference frame (time dilation)and the observer is not.
Simplistically, maybe, does time dilation stem from the act of measurement (light having to travel a longer distance, etc. if the clock is moving relative to the observer) happening from a frame that is not at rest with the clock?
If the measurement act was not involved, would the clocks on different inertial frames moving w.r.t. each other tick the same way, with the same rhythm?

#### PeterDonis

Mentor
If the measurement act was not involved, would the clocks on different inertial frames moving w.r.t. each other tick the same way, with the same rhythm?
No. You are making way too much of the word "measurement" here. A clock ticks time. That does not mean its ticking has to be viewed as "measuring" time, or that you can somehow still have the ticking while taking away the "measuring". The "measuring" is the ticking, because that's all the clock is doing.

To understand how a light clock in motion seems time dilated, think about the case of two light clocks in relative motion, both of which tick at the same rate in their respective rest frames (i.e., in their respective rest frames, the clocks are geometrically identical, same mirror spacing). Relative to either clock, the other clock will appear to be ticking slow. No "measurement" needs to be involved other than the comparison of the ticks.

• nasu

#### fog37

I see. Thanks. Sometimes it seems like the time dilation derives from the fact that there are photons that need to travel at a fixed speed from the clock to an observer in motion and that process of transmitting the time information from the clock to the observer causes the time measurement to "appear" dilated. But as you, mention, the time interval is truly dilated for a clock in motion (ticks more slowly)...

#### PeterDonis

Mentor
Sometimes it seems like the time dilation derives from the fact that there are photons that need to travel at a fixed speed from the clock to an observer in motion and that process of transmitting the time information from the clock to the observer causes the time measurement to "appear" dilated.
There is no such process anywhere in the light clock scenario. There is only the ticking of the clock.

#### fog37

Ok, thanks. But to read an ordinary mechanical and moving clock, doesn't light from the clock have to travel from the clock to the observer?

#### PeterDonis

Mentor
to read an ordinary mechanical and moving clock, doesn't light from the clock have to travel from the clock to the observer?
Not if the observer is standing next to the clock. (Strictly speaking, the observer has to see the clock so light does travel from the clock to the observer's eyes, but this travel time is negligible if the observer is standing next to the clock.)

Allowing for light travel time for observers not standing next to an object being observed is a completely separate issue.

#### Ibix

Ok, thanks. But to read an ordinary mechanical and moving clock, doesn't light from the clock have to travel from the clock to the observer?
To read any clock, light or otherwise, a signal has to travel from that clock to the observer. But time dilation is what is left after you correct for that travel time. And you can easily correct for it by having multiple clocks.

For example, have three clocks in a row, stationary with respect to one another and synchronised. Each one has an observer colocated with it. To make the maths simple, space them a light second apart (incidentally, this means that when the first clock reads 12:00:00 an observer co-located with it will see the second clock read 11:59:59 and the third 11:59:58 - but can see that the offset is constant and can use radar to cinvince himself that this is expected behaviour of synchronised clocks). Now a fourth clock moves past them at 0.8c (producing a time dilation factor of 0.6). As it passes each stationary clock, the observer at that clock notes down the time on their stationary clock and the moving clock, both from arbitrarily short range. When they get together and compare their readings, they will find that their stationary clocks show that the second stationary clock time was 1.25s after the first and the third was 1.25s after that - consistent with the travelling clock taking 1.25s to cross one light second at 0.8c. The travelling clock readings, however, will have advanced by only 0.75s each time due to time dilation.

Again, note that light travel time is negligible here, and that I have not specified anything about the type of clock used.

#### A.T.

Question: is the path followed by the light pulse straight or zig-zag?
Even in Galilean Relativity the path shape is different in different frames.

Sometimes it seems like the time dilation derives from the fact that there are photons that need to travel at a fixed speed from the clock to an observer...
No, that has nothing to do with time dilation. The misconception comes from using the word "observer" for "reference frame", which implies a certain position.

#### fog37

Thank you everyone!

In the example of the light clock: we start with two identical light clocks, one on the ground (frame $A$) and one in a car moving at constant velocity (frame $B$).

According to observer $A$ in frame $A$, the rate of ticking of of the light clock is such that a measured time interval is the proper time $\Delta t_{proper}$, which is the difference between the times $t_2$ and $t_1$ of two events happening at the same spatial position (point of emission=point of detection of the light flashes).

However, when observer $A$ measures the time interval on the identical clock that is inside the car, frame $B$, the measured time is different and equal to $\Delta t_{dilated}$>$\Delta t_{proper}$, correct?

The dilated time $\Delta t_{dilated}$ measured by observer $A$ is the time interval measured by the clock inside the moving car from the reference point of observer $A$ in frame $A$. To carry out this time interval measurement, frame $A$ is to be imagined as a lattice of synchronized clocks located at every point in space. So when the car and the clock inside it pass by these frame $A$ clocks (i.e. at rest with frame $A$), measurements are taken and a $\Delta t$ is calculated which happens to be $\Delta t_{dilated}$... Is that correct? That is how time intervals or position differences (lengths) are measured from one frame when the object of interest is moving relative to it.
1. On a different notes, in SR the "speed" of light is always constant... but what about the "velocity" of light? Are there situations in which the direction is not?
2. In the muons example, where muons make it to earth in abundance, the phenomenon can be explained with both time dilation or length contraction, correct? Is one explanation more correct than the other? Or are time dilation and length contraction "related" concepts or even the same concept in some way? I don't grasp their connection yet...
Thank you!!!

#### Nugatory

Mentor
In the muons example, where muons make it to earth in abundance, the phenomenon can be explained with both time dilation or length contraction, correct? Is one explanation more correct than the other? Or are time dilation and length contraction "related" concepts or even the same concept in some way? I don't grasp their connection yet...
The two explanations are different ways of saying the same thing, namely that the proper time between the events (points in spacetime) “muon is created” and “trajectory of muon intersects surface of earth” is less than the muon lifetime so the muons make it to the surface of the earth.

#### A.T.

In the muons example, where muons make it to earth in abundance, the phenomenon can be explained with both time dilation or length contraction, correct? Is one explanation more correct than the other? Or are time dilation and length contraction "related" concepts or even the same concept in some way?
See the other thread:
When starting to learn relativity I would strongly recommend not using the length contraction or time dilation formulas. You should stick exclusively with the Lorentz transform equations until you are more advanced. It is too easy to misuse the length contraction and time dilation formulas.

#### Ibix

In the example of the light clock: we start with two identical light clocks, one on the ground (frame $A$) and one in a car moving at constant velocity (frame $B$).

According to observer $A$ in frame $A$, the rate of ticking of of the light clock is such that a measured time interval is the proper time $\Delta t_{proper}$, which is the difference between the times $t_2$ and $t_1$ of two events happening at the same spatial position (point of emission=point of detection of the light flashes).

However, when observer $A$ measures the time interval on the identical clock that is inside the car, frame $B$, the measured time is different and equal to $\Delta t_{dilated}$>$\Delta t_{proper}$, correct?
It isn't clear to me what time interval you are measuring on which of the two clocks you started with. You need to specify a start and a stop event and say whose clocks we are using in any given measurement.
The dilated time $\Delta t_{dilated}$ measured by observer $A$ is the time interval measured by the clock inside the moving car from the reference point of observer $A$ in frame $A$. To carry out this time interval measurement, frame $A$ is to be imagined as a lattice of synchronized clocks located at every point in space. So when the car and the clock inside it pass by these frame $A$ clocks (i.e. at rest with frame $A$), measurements are taken and a $\Delta t$ is calculated which happens to be $\Delta t_{dilated}$... Is that correct? That is how time intervals or position differences (lengths) are measured from one frame when the object of interest is moving relative to it.
Again, it is not clear what you are measuring. You can conceptualise a frame as a grid of synchronised clocks, yes. Then you always have a local clock for any event and you just read that one, if that's what you mean. It doesn't "happen to be" equal to the dilated time - the ratio of interval measurements made this way to the proper time between events is how time dilation is defined.

There are other ways of making time measurements (using a single clock and correcting its readings for the lightspeed delay, for example), but they are equivalent to the above.
On a different notes, in SR the "speed" of light is always constant... but what about the "velocity" of light? Are there situations in which the direction is not?
Google "relativistic aberration". The underlying phenomenon as the same one that makes vertically falling raindrops make diagonal streaks on your car's side windows.
In the muons example, where muons make it to earth in abundance, the phenomenon can be explained with both time dilation or length contraction, correct? Is one explanation more correct than the other?
No. They are two ways of describing the same thing. In the Earth frame the muons' half life is increased due to their time dilation, so they make it to Earth (rulers attached to the muons would also be length contracted in this frame, but that is irrelevant for this example). In the muon frame, their half lives are very short but the atmosphere is very shallow due to length contraction so they don't have to survive very long before being hit by the Earth (clocks on the Earth would also be time dilates in this frame, but that is irrelevant for this example).
Or are time dilation and length contraction "related" concepts or even the same concept in some way? I don't grasp their connection yet...
As A.T. says, it is far better to forget about time dilation and length contraction and focus on the full Lorentz transforms. The time dilation and length contraction formulae are special cases of the Lorentz transforms, and it is very easy to apply them in situations where they don't work and end up very confused. Once the Lorentz transforms are second nature the answer to your question will be obvious.

#### fog37

Thank you! I have studied and learnt the Lorentz transformations today and now see how the two effects (time dilation and length contraction) neatly arise from them.

As far as SR first postulate, which says that the "laws of physics are the same in all inertial frames"...
What does it really mean? For example, let's talk about Newton's law, $F_{net}=ma_{net}=F_1 + F_2+F_2+etc$. All the listed forces are real forces acting on the object. If the reference frame was not inertial, we can still use Newton's law but we need to introduce fictitious, fake, forces (strangely called inertial even if they pertain to noninertial frames).
But the same physical situation is mathematically described from two different inertial frames with an identical equation (of motion if we are concerned about motion) $F_{net}=ma_{net}=F_1 + F_2+F_2+...$, including exactly the same terms, i.e. the same forces, even if the velocities, the position coordinates are different. The lengths and time intervals are also the same in Newtonian physics but possibly not the same in relativistic physics...Is that correct?
I guess Maxwell's four equations, before SR, had different forms in different inertial frames and that was a problem...

#### fog37

I was reading that Newton's laws are "covariant" to Galileian transformation but Maxwell's equations are not.
However, both Newton's and Maxwell's equations are covariant w.r.t. Lorentz transformations.

I assume covariant means that they are exactly the same....

#### PeterDonis

Mentor
both Newton's and Maxwell's equations are covariant w.r.t. Lorentz transformations.
No, that's not correct. Maxwell's Equations are Lorentz covariant. Newton's laws are not. They can't be, because no set of equations can be covariant to both Galilean and Lorentz transformations.

I assume covariant means that they are exactly the same....
It means the equations take the same form when transformed to a new frame using the specified transformations (Galilean or Lorentz).

#### fog37

Thanks PeterDonis.

The word "covariant" means changing with...shouldn't equations that do not change w.r.t. Lorentz transformations be called Lorentz invariant?

So, Newton's laws are not wrong, they just work for motion at low speeds.

SR 2nd postulate talks about "ALL laws of physics are the same in any inertial frame". I guess it refers to all laws except some?

#### Ibix

So, Newton's laws are not wrong, they just work for motion at low speeds.
To some extent they are inaccurate at any speed. It's just that you need impossibly precise measurements to detect the error from using them at low speed - they are good enough.
SR 2nd postulate talks about "ALL laws of physics are the same in any inertial frame". I guess it refers to all laws except
No - all laws of physics are Lorentz invariant, so far as we are aware. Newton's laws are only approximations (really, really good ones often) to relativistic and quantum physics.

• Dale

#### PeterDonis

Mentor
The word "covariant" means changing with...shouldn't equations that do not change w.r.t. Lorentz transformations be called Lorentz invariant?
Often "Lorentz invariant" is indeed the term that is used. Don't get too hung up on "covariant" vs. "invariant". The key point is the math, not the words used to describe it.

• fog37

#### fog37

Thank you Ibix.

Beside Maxwell's four equations, which other equations are Lorentz invariant?

And why the word "covariant"? Covariant in what sense? I know there is also the term contravariant...

#### Dale

Mentor
SR 2nd postulate talks about "ALL laws of physics are the same in any inertial frame". I guess it refers to all laws except some?
No, it refers to all laws of physics. The question is whether a given equation is a law of physics or an approximation. Newton’s laws are not laws of physics, they are low speed approximations to relativistic laws.

• fog37

#### Nugatory

Mentor
And why the word "covariant"? Covariant in what sense? I know there is also the term contravariant...
Components of tensors can be described as either “covariant” or “contravariant”, but that’s a completely different use of the word than in this thread. It is unfortunate that the same word is used to mean different things, but that’s just the way the English language evolved - the context will make it clear which meaning is intended.

• Ibix and fog37

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