# Effect of Relativity on Laser Sail Acceleration

• I
• Sebastiaan

#### Sebastiaan

Question, does general or special relativity have any effect on the acceleration of laser sail from an external observer? Let's say a laser-sail is first accelerated to 0.2c. Ignoring diffraction, Due to redshift, the incoming laser beam would get stretched by 20%, reducing the power by 20%. Beside the 20% reduction in acceleration, would general or special relativity have any additional effect?

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Yes, but not by much at this speed yet. Let's assume that we were able to keep increasing the output of the laser so that, as measured by the light sail, its acceleration remains constant. As measured by the Earth, the acceleration of the light sail would decrease as its velocity increased. Imperceptibly at first, but more and more noticeably as it speed got closer to c.

It does? Then let's say the lasersail is accelerated to 0.9c, it would receive 90% less power from red shifting alone. Now exactly how to calculate the continued acceleration of the laser as a percentage of maximum acceleration (at start).

Here is my attempt:

Time Dialation = 1 / Sqrt[ 1 - (v2 / c2) ] = 2.294157

Acceleration = (1 - 0.9) / 2.294157 = 0.04358899 = 4.359%

Correct?

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Question, does general or special relativity have any effect on the acceleration of laser sail from an external observer? Let's say a laser-sail is first accelerated to 0.2c. Ignoring diffraction, Due to redshift, the incoming laser beam would get stretched by 20%, reducing the power by 20%. Beside the 20% reduction in acceleration, would general or special relativity have any additional effect?

The amount of energy (and momentum) in the laser beam varies as ##1 / \gamma^2##, where ##\gamma = 1/\sqrt{1-\left( \frac{v}{c} \right)^2}. ## , and I suspect ##\gamma## is what you mean by the "stretch factor". But see below. ##1/\gamma^2## works out to be more simply expressed as ##\left(1 - \left( \frac{v}{c} \right)^2 \right)##

However, at .2c, the "stretch factor" ##\gamma## is only 2 percent, not 20 percent. We can see from the formula that the acceleration drops by 4 percent, that it is .96 of the acceleration without relativistic effects.

One way of understanding the effect is inverse relativistic beaming, https://en.wikipedia.org/wiki/Relativistic_beaming

It's convenient to look at it in the photon viewpoint as well, though of course since it's purely classical we don't need to do this. The energy and momentum in each photon drops by a factor of ##1/\gamma## due to the frequency shift, and the number of photons per second, the intensity, also drops by a factor of ##1/\gamma##, so the net effect on the laser beam as seen in the frame of the moving spaceship is that its power and momentum/second drop by a factor of ##1/\gamma^2##.

However, at .2c, the "stretch factor" ##\gamma## is only 2 percent, not 20 percent. We can see from the formula that the acceleration drops by 4 percent, that it is .96 of the acceleration without relativistic effects.
Ok, now I'm confused, so the standard doppler effect math for light can be ignored and instead we only calculate 1/ ##\gamma##2 ?

edit: according to the dutch wikipedia, the doppler effect for light can be calculated with which would result in a 10.557% acceleration reduction

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I'm not quite sure what you mean by "effects of relativity". My guess is that you are asking what different predictions would relativity make compared to Newtonian physics.

I would say (if I understand what @pervect is doing, he disagrees) the easy way to work this problem is in the initial rest frame of the sail. The energy striking the sail per unit time when it's traveling at velocity v is then exactly the same in the relativistic case as the Newtonian case. What's different is how the sail responds. Its kinetic energy is ##\gamma mc^2## and its momentum is ##\gamma mv##. At low speed these are approximately the same as the Newtonian formula, but as you approach light speed they grow without bound, so the velocity increase from a given amount of laser energy reduces towards zero.

You don't need to bring length contraction or time dilation into this until you want to work out what the passengers on the light sail see. Qualitatively, you see reducing proper acceleration, probably with a profile different from the Newtonian case, but I don't know for certain.

I would say (if I understand what @pervect is doing, he disagrees) the easy way to work this problem is in the initial rest frame of the sail. The energy striking the sail per unit time when it's traveling at velocity v is then exactly the same in the relativistic case as the Newtonian case. What's different is how the sail responds. Its kinetic energy is ##\gamma mc^2## and its momentum is ##\gamma mv##. At low speed these are approximately the same as the Newtonian formula, but as you approach light speed they grow without bound, so the velocity increase from a given amount of laser energy reduces towards zero.
So if I understand you correctly you first calculate the basic Newton dopler effect and then adjust for time dilation meaning a laser sail traveling at 0.2c, would from the initial frame of reference (at beam station) be observed to travel at an acceleration of 0.8 * 1.0002 = 0.800016 of its initial acceleration (assuming no diffrraction and transmitted beamed power would remain constant)

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I don't know what the 0.8 or 1.0002 are supposed to be. Where did they come from?

I find this easiest to conceptualise in terms of pulses of laser light striking the mirror. Regular pulses of light strike the sail at longer intervals when it's moving. So the power striking the sail decreases. Furthermore, the energy the sail derives from each pulse changes - each incident pulse has the same energy, but the energy it carries away once it's reflected changes.

So there are two sources of change in the rate of change of kinetic energy of the sail. Did you account for both? Once you have that change in kinetic energy you can write down the rate of change of velocity easily.

Yes I think so,
the 0.8 comes from Doppler effect traveling at 0.2c : 1 - (1 - 0.8 / 1)
the 1.0002 is the time dilation effect when traveling at 0.2c :1 / Sqrt[ 1 - (v2 / c2) ]

so when combined acceleration would be 0.8 / 1.0002 = 0.7998 of initial acceleration

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the 1.0002 is the time dilation effect when traveling at 0.2c :1 / Sqrt[ 1 - (v2 / c2) ]

Then the increase factor of the obsolete longitudinal relativistic mass is 1.00023.

And the decrease factor of the force was the basic Newton Doppler shift factor squared.

Now we can calculate the acceleration a=F/m.

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Ok, now I'm confused, so the standard doppler effect math for light can be ignored and instead we only calculate 1/ ##\gamma##2 ?

No. My bad. Please ignore what I said earlier - I was working from memory, and it played some tricks on me. (Plus, I didn't understand what you meant by the stretch factor, know I know you mean the doppler factor, which is ##\gamma (1 - \beta)##. Andthat is 20 percent.

I did work this out once upon a time, though I was mainly interested in the ultra-relativistic limit. I'll review some of my earlier work and try to post something more careful and hopefully correct next time around.

• Sebastiaan
I found the post I was thinking of, https://www.physicsforums.com/threads/photon-arrival-rate.681172/#post-4323336

The non-technical argument from that post goes like this:

If you have n photons in one wavelength of the beam, then when you boost the wavelength grows by your "stretch factor" (which I called the doppler factor in the original post). The number of photons in one wavelength stays the same, that's something that is observer independent in special relativity. This implies that the number of photons in a unit length goes down by a factor of the stretch factor. The energy (and momentum, since E=pc) of each photon also goes down by the stretch factor. So in one second, the number of photons hitting the light sail will be the number of photons in a unit length of 1 light second. Since we've just argued that the number of photons per unit length drops by the stretch factor, the number of photons per second hitting the light sail drops by the stretch factor. The momentum carried by each photon also drops because the frequency drops and E= pc = h c / ##\lambda##. The end result is you get a reduction of (1.2)^2 in your thrust for a stretch factor of 1.2.

• Sebastiaan
The momentum carried by each photon also drops because the frequency drops and E= pc = h c / ##\lambda##. The end result is you get a reduction of (1.2)^2 in your thrust for a stretch factor of 1.2.
So the Doppler redshifting effect alone is responsible for a reduction of 44% in acceleration, that's brutal! Now the remaining question is, besides the Doppler effect, does it also experience a time dilation effect as the vessel mass would appear to increase. It may not be much, but it is still relevant.

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So the Doppler redshifting effect alone is responsible for a reduction of 44% in acceleration, that's brutal! Now the remaining question is, besides the Doppler effect, does it also experience a time dilation effect as the vessel mass would appear to increase. It may not be much, but it is still relevant.

The way I'm structuring the calculation, there's no need to take into account time dilation, as I'm calculating the proper acceleration of the sail, which is the acceleration of the sail in its own (instantaneous) inertial frame.

This acceleration would go into the "relativistic rocket equation", for instance. http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

• Sebastiaan
Alright, so we now calculate the proper acceleration of the LightSail when it is moving straight away from it source, but what happens it is not going straightaway but at 45 Degree angle (to make some coerce corrections), will acceleration still be modified by (v/c)^2 or less (because we are effectively move slower away from the source)?

Alright, so we now calculate the proper acceleration of the LightSail when it is moving straight away from it source, but what happens it is not going straightaway but at 45 Degree angle (to make some coerce corrections), will acceleration still be modified by (v/c)^2 or less (because we are effectively move slower away from the source)?

You just calculate the frequency and intensity of the light in the spaceship frame, using the relativistic doppler formula, find the momentum in the light from the frequency and intensity, and then apply the conservation of momentum to get the net thrust. If the light isn't being reflected straight back at the source, one does a vector subtraction of the incoming light momentum minus the outgoing light momentum to get the net momentum transferred to the sail. So there are multiple effects, one due to the doppler effect (as discussed in the wiki article https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Transverse_Doppler_effect for the transverse and angled case), and another effect due to the angle of the sail if the sail isn't reflecting the light back at the source. Twice the momentum of the light is transferred to the spaceship when the light is reflected straight backwards, this is lowered if the sail is angled.