Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: katieHaylor on 26/06/2017 21:28:00
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Donald says:
Do gravitational waves undergo Doppler shift like light and sound waves? And how could it be measured?
What do you think?
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Inertial reference frames use the Lorentz (time) Transformation show below from Relativity:
t' = γ (t - v x / c²)
Using the above we can directly derive the Relativistic Doppler Shift below:
f'/f = γ (1 - v / c) (the connection is f'/f = t'/t)
Note: when γ is about 1 ( v << c ) the above matches the Classical Doppler Shift
Inertial reference frames shift via the Lorentz Transformation. Likewise, Anything moving at the speed of light that has a frequency shifts via the Relativistic Doppler Shift.
We could measure the Doppler Shift by having gravitational wave detectors moving at different speeds relative to each other. Say one is on Earth and one is in space. We could then detect a shift between the results of the two detectors.
Gravitational waves should also be stretched out as the universe expands. More distant events should be more red shifted.
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The gravitational waves from a black hole merger 1 billion light-years away would be Doppler-shifted by the general expansion of the universe.
The sorts of gravitational wave frequencies we can detect with LIGO have been around 50-400Hz.
The most differential velocity we could easily get in the Solar system is to place a space-based gravitational wave detector in a similar orbit to the Earth, but at (say) 120 degrees in front and behind Earth's location. This would give us great time resolution on the time of arrival of the gravitational waves, allowing a precise location in the sky.
But orbital velocity differences of 30 kilometers/second or so would produce a very small frequency shift on a signal with a frequency of 50-400Hz.
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There should generally be a ratio of around 50:1 between the frequencies of gravitation and electromagnetism over comparable ranges of frequencies based on a comparison of both spectrums. At the moment I don't have access to my notes so can't elaborate.
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There is also the consideration of the different effects of gravitational lensing on electromagnetic versus gravitational waves.
http://iopscience.iop.org/article/10.3847/1538-4357/835/1/103/meta
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But orbital velocity differences of 30 kilometers/second or so would produce a very small frequency shift on a signal with a frequency of 50-400Hz.
With a speed difference of 30,000 m/s the shift should be about z = | f' - f | / f ≈ v/c
v/c = 30,000/300,000,000 = 1/10,000 or one-part in ten thousand.
Gravitational waves we detect ramp up and last on the order of about a second. If the event lasts a second on Earth I would think the event could be seen as blue or red shifted by up to about 1/10 of a millisecond in the space based detector.
Couldn't we measure the relative duration of the entire event and compare? I would think we'd be able to do this but I don't know how hard this would be to measure. We have clocks much more accurate than 1/10 of a milisecond but then again gravity waves are already so hard to detect.
The value z for an event about 1 billion light-years away is about .1 or 1,000 times larger than the Doppler Shift due to the speed difference of 30 km/s.
I used the following:
The Hubble Constant is 100km/Mps and a Mps = 3.26 million light-years
1,000 / 3.26 ≈ 300
300 * 100,000 m/s = 30,000,000 m/s
z ≈ v/c = 30,000,000 / 300,000,000 = .1 (γ still isn't that much different with v = .1c)
If a gravity wave lasts 1 second for us on Earth but the event occurred 1 billion light-years away that event should have been about .9 seconds close to where it occurred (say within the galaxy it occurred). This is a rough estimate and doesn't include the drift velocity which varies.
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Couldn't we measure the relative duration of the entire event and compare?
The time taken for black holes to merge is probably of the order of millions of years, slowly spiralling closer together. But the strongest gravitational waves are radiated in just the final few milliseconds.
If we take the first gravitational wave detection at LIGO, the duration of the detected event lasted about 16ms.
This is from the time that the signal emerged from the noise floor with a frequency around 50Hz, hit a peak at around 400Hz, and then disappeared back into the noise floor with a frequency around 100Hz. The "cleaned-up" graphs in the middle have been filtered to remove noise that didn't fit with the chirp of a black hole merger.
It is very hard to measure the frequency to one part in ten thousand when you only detect 13 cycles of the signal, the actual frequency changes rapidly over those 13 cycles, and the signal is not far above the noise floor.
See: https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves
...expand the graphs on the right.
However, a space-based telescope would be much more sensitive, measuring over a baseline of perhaps a million km (instead of 4 km). So perhaps it could detect and separate out multiple sources over a period of years when the frequency is relatively stable. This may allow separating out Doppler shift.
See: https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna
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For the duration I was going off of this timing I found below and graphs I've seen.
The black holes were smaller than in the first detection event, which led to different timing for the final orbits and allowed LIGO to see more of the last stages before the black holes merged—55 cycles (27 orbits) over one second, with frequency increasing from 35 to 450 Hz, compared with only ten cycles over 0.2 second in the first event.[1][5]
It definitely makes sense that details get lost in the noise when considering the image below (I know they use powerful computers to sort through the data and to compare the signals from the two different detectors in the different locations):
https://en.wikipedia.org/wiki/GW170104#/media/File:GW170104_signal.png
It also makes sense that we only first detected the last few milliseconds and work back from there. I guess well have to see how good the detectors get.