by: David Butler

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at the end of the second segment on general relativity I pointed out that Einsteins prediction that accelerating masses can create gravitational waves had yet to be discovered as soon as I released the book they were discovered so I have added this chapter to the video book to go into gravitational waves what they are what is a ripple in space-time what creates them what can create a gravitational wave large enough for us to detect and how did we actually detect it now we used interferometers the Michelson interferometer to actually detect the first gravitational wave I'll end by discussing the impact on cosmology that the discovery of gravitational waves means but first what is a ripple in space-time here on earth far from an event that could create a gravitational wave we have a relatively flat space with Euclidean metric G that isn't changing with time a ripple represents small deviations from this flat space-time metric we use H to represent these deviations solutions to Einstein's equations show that a gravitational waves metric oscillates sinusoidal e just like light and it travels at the same speed as light as a wave moves down the z axis planes at different times experience different values for the metric used to measure distance on the plane this makes the wave a transfer sway just like light we see two possible polarizations for a gravitational wave we call one h+ for the action along the x and y axis we call the other h cross for action along the diagonal to see what an oscillating H+ metric does we'll measure the changes in the distance between two points on the plane when a gravitational wave passes here we have an XY plane with the wave passing into the page we mark two points on the x axis one meter apart in Euclidean flat space where H is zero when H is greater than zero the distance between the two points on the x axis become longer than one meter by an amount equal to H times the original distance at the same time a one meter distance on the y axis will shrink to less than one meter by the same amount when H returns to zero the distance between these points returns to one meter when H is less than zero the distance between the two points on the x-axis will become shorter than 1 meter and the distance between the two points in the y-axis will become longer than 1 meter here's an exaggerated look at what an oscillating h plus polarized gravitational wave does to a square plate it passes through again the wave is passing into the page for an H cross polarized wave the effect would be similar but shifted 45 degrees when describing a gravitational wave we can now be more precise than its a ripple in space-time a gravitational wave is an oscillating polarized metric that operates in the plane perpendicular to the direction of the wave as it moves through space at the speed of light and we have seen what this means for the objects that encounter such a wave they are stretched and squeezed in various directions will now turn our attention to the kinds of massive accelerating objects that can create such a wave in order to generate a gravitational wave you need a non spherically symmetric rotating system for example here's a binary star system with two masses revolving in a circular orbit around a common center of gravity the Stars acceleration creates gravitational radiation that travels out from the system in all directions just like the light they are generating the gravitational wave solutions show that the frequency of the created gravitational wave is twice the rotation rate of the binary system we also see that the polarization and maximum gravitational wave amplitude depend on the masses of the two objects the distance between them their rotational velocity the viewing angle and how far away the system is from the observer

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there is one more key factor to consider when it comes to binary systems namely that the gravitational waves carry energy and momentum away from the system we call as gravitational luminosity Newton and Kepler provided the mechanics for understanding what happens to the orbit when gravitational energy is lost because binding energy is negative a loss of energy will make it a larger negative this has the effect of reducing the distance between the two objects this in turn increases their velocity a shorter circumference and faster velocity reduces the time it takes for a full orbit and therefore increases the frequency of rotation and therefore the frequency of the gravitational wave and the wave equations show that the amplitude of the gravitational wave will increase with the frequency the rate that the frequency is changing is called the chirp it gives us the ability to express the amplitude of the gravitational wave in terms of the frequency and the rate the frequency is changing instead of the masses and the distance between the masses this is crucial because for most cases we will have no way of knowing directly what the masses are or how far apart they are but measuring the frequencies might be possible if we can also measure the amplitude we can even calculate the distance to the binary system because this distance is based on gravitational wave luminosity it is called the luminosity distance for most all gravitational wave sources this will be the only way to figure out how far away they are

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with the decaying orbit the objects will eventually collide and coalesce the resulting wave form call the coalescing wave form serves as a signature for this kind of gravitational wave source it has three phases the in spiral the merger and the ring down to an object that is no longer asymmetric and therefore no longer radiating gravitational waves to get an idea on the expected amplitudes and frequencies for gravitational waves created by a system like this one let's put in some numbers suppose the system is a hundred light years away and each star is the mass and size of our Sun at the point where they are about to touch we would see the maximum amplitude in this example we get 10 to the minus 21 this is a very small number it is approximately the ratio of the width of the human hair to the distance to Alpha Centauri for light-years away here is where this data point fits on a graph with wavelength decreasing along the x axis and amplitude increasing along the y axis binary systems like this one are plentiful and all around us there are literally billions of them sending gravitational waves our way from every direction but the gravitational waves they create or weak and totally indistinguishable from one another they just wind up contributing to a background noise level in our sensitivity graph we see that in order to detect a gravitational wave a binary system will have to create waves with greater amplitudes and higher frequencies to generate smaller wavelengths than the noise level marked in green to stand out a binary system is needed they can achieve much higher velocities and as we have seen from our example the large diameters of stars prevents them from ever getting close enough to reach the needed velocities but rotating neutron stars might be small enough to achieve the needed speed here's a system with two equal mass neutron stars that have reached the point where they are whirling around each other ten thousand times a second the Stars merge in a few milliseconds sending out a burst of gravitational waves and a brief intense gamma-ray burst you can see the three phases the in spiral the coalescence or merger and the ring down to an object most likely a kerr black hole that is no longer asymmetric and therefore no longer radiating gravitational waves if we fed the wave form into an audio generator it would sound like this we call it the chirp the mass of a typical neutron star is 1.5 times the mass of the Sun with a radius of only 10 kilometers if the system is 33,000 light-years away an average distance for a Milky Way object it would give us a theoretically detectable wavelength and amplitude but coalescing neutron stars are not common events astronomers estimate that there might be one of these neutron star mergers every 50 years inside the Milky Way to get a higher rate we have to move outside of the galaxy into the Virgo supercluster our local supercluster that we covered and how far away is a video book within a 50 million light years we expect to have as many as 10 or more neutron star mergers per year because we're including thousands of galaxies unfortunately at this extended distance the amplitude drops to the 10 to the minus 21 range stellar-mass black holes can get as close as neutron stars because their swartz child radius is larger than the radius of a neutron star but their mass alone can create larger gravitational wave amplitudes here we see a black hole merger simulation if each black hole has a mass of seven Suns the swartz child radius is 20 kilometers twice the radius of the neutron stars as the orbital radius shrinks to twice the Schwarzschild radius and the black holes are approaching each other's photosphere their velocities approach 70% of the speed of light this produces a shorter gravitational wave length and a larger amplitude putting this kind of event well into the theoretically detectable area a number of other major cosmological events can also create gravitational waves here is a chart of some of the events and their expected wavelengths and amplitudes supernovae binary mergers like the ones we've been analyzing supermassive black hole mergers and remnants from the Big Bang in 1974 58 years after Einstein predicted the existence of gravitational waves two radio astronomers Joe Taylor and Russell Holtz we're looking for new pulsars using the 305 meter Arecibo radio telescope in Puerto Rico they found one it's named PSR be nineteen thirteen plus sixteen and it led to the first indirect verification of Einstein's prediction you'll recall from the globular clusters and supernovae chapter in the how far away is a video book that a pulsar is a rapidly rotating neutron star with a powerful magnetic field the result is a sort of magnetic lighthouse which if aligned correctly flashes in our direction twice each cycle these signals are highly regular in fact pulsars are some of the best clocks in nature and this allows extremely precise measurements of their motion this one was pulsing every 59 milliseconds indicating that the Pulsar rotates 17 times per second but Holtz and Taylor noticed that the pulsars varied regularly every 7 and 3/4 hours with pulses arriving three seconds earlier at some times relative to others this meant that the pulsar was in an elliptical orbit with another neutron star this was the first binary neutron star ever discovered using the orbital motion they calculated the star's masses their closest approach call a paris tron and their farthest distance apart call e-apis tron as well as the system's inclination with this information and the gravitational wave equations they were able to calculate the amount of gravitational radiation the expected decay of the orbit due to the lost gravitational energy and the corresponding reduction in the time it takes per orbit this graph maps the accumulated reduction in orbital periods against time assuming that Einstein's equations are correct ult's Taylor and others have studied this binary system for 40 years now this graph records their measurements we see that the measurements fit the theory perfectly this gave scientists confidence that Einstein's gravitational waves do indeed exist but direct mein tricky for two main reasons one is that the amplitude of the waves are so small and the other is that the measuring sticks you might use to measure a change in length are changed themselves in other words the change that lengths will still read out as one meter but the stretching and squeezing does put a strain on the plate and that can be measured with an attached wire that acts as a resistor it's called a strain gauge if we attach wires along the plate instead of a meter stick we can measure changes in the resistance of the wire as it is stretched and squeezed a longer thinner wire will provide more resistance to an electric current and a shorter fatter wire will provide less resistance to an electric current that's giving us a measure of the strain unfortunately this technique is literally millions of times too insensitive to measure the tiny gravitational wave amplitudes H but this technique is why we call H a measure of strain Michelson interferometer x' look like the best chance to detect these waves you recall that we covered the interferometers in the first chapter of this video book the arms on that one were 11 meters long and its sensitivity was nowhere near what is needed for gravitational waves today we have LIGO the laser interferometer gravitational-wave Observatory that has built two identical interferometers 3000 kilometers apart with one near Hartford Washington and the other near Livingston Louisiana here are the l-shaped Lego instrument components it has a powerful near infrared laser with an output after amplification that reaches 200 watts of 1064 nanometer light the beam splitter and mirrors that act as test masses are 40 kilogram objects suspended by a fused silica glass fibers to minimize noise due to vibrations additional internal and external active vibration minimization technologies eliminate the effects of everything from nearby traffic to lunar tidal forces the four kilometer arms are ten thousand cubic centimeters of ultra high vacuum equal to one trillionth of an atmosphere in addition each arm contains reflective mirrors that are out the light back and forth inside the arms 280 times before it hits the exits for recombination the photo detector is a state-of-the-art indium gallium arsenide photodiode array with a high quantum efficiency designed to detect extremely small amounts of light at a wavelength of 1064 nanometers the laser light is split and sent to the two mirrors on return they are recombined and sent to the photo detector the beams returning from the two arms are kept out of phase so that when the arms are both in sync as when there is no gravitational wave passing through their light waves subtract and no light arrives at the photo detector when a gravitational wave passes through the interferometer the distance along the arms of the interferometer are shortened and lengthened causing the beams to become slightly out of sync and sunlight arrive to the photo detector indicating a signal given LEGOs extra 280 passes through the tube a gravitational wave strain amplitude of 10 to the minus 21 would displace the mirrors by 10 to the minus 18 that's one thousandth the diameter of a proton on our sensitivity graph we see where Legos characteristics fit this is a range or powerful binary system mergers within the Virgo supercluster should be detectable at 950 and 45 seconds coordinated Universal Time on the 14th of September 2015 a signal was detected by the LIGO detector in Livingston and 6.9 milliseconds later in Hanford it was a chirp signal that lasted just over two tenths of a second when we route the wave into a sound generator here's what it sounds like

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this plot combines the data from both sites the waveform is consistent with coalescing masses with a 10 cycle 200 millisecond Inspiral that gives us the frequency the rate of change of the frequency and a peak wave amplitude a merger that takes around 2 milliseconds and a ring down as the coalesced objects cease to radiate gravitational energy detector noise introduces errors into all the calculations based on these figures that's why we'll provide a range for each item the amplitude and frequency data points give us the luminosity distance it is important to note the gravitational waves experienced redshifting as they travel across the cosmos just like light does having traveled around a billion light years this wave would have experienced a redshift near 0.1 so the frequency we see here is a bit smaller than the frequency at the start of the waves journey here the frequency data also gives us the chirp mass taking the redshift information gleaned from the merger and ring down portions of the wave form we get the binary system masses these masses are too large for neutron stars that are only a few times the mass of the Sun so we must be witnessing the merger of two large stellar black holes

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during the last 200 milliseconds of their in spiral the orbiting velocity of the black holes increased from 30 percent the speed of light to 60 percent of the speed of light over the same period the distance between the two black holes went from around a thousand kilometers to just under 200 kilometers when their event horizons made contact modeling the final ring down shows that the mass of the resulting curb Lac hole is around 62 solar masses that's three solar masses less than the sum of the masses of the two in spiraling black holes this mass was converted to the radiated gravitational energy in other words during the final 20 milliseconds of the merger the power of the radiated gravitational waves peaked at about three point six times ten to the 49th watts let's take a second to get a feel for how large this number is in our how far away is its segment on nearby stars we found that the Sun converts four point two six metric tons of matter into energy every second the resulting power output is equal to four billion hydrogen bombs exploding every second the Sun is an average star so we can use this as an average stellar power output from our segment on local superclusters we saw that there are 250,000 trillion stars within 1 billion light years is represented around 7 percent of the total number of stars in the universe we get the total power emitted by all the stars in the visible universe by multiplying the average watts per star times the number of stars the power generated by this merger of the two stellar-mass black holes is 26 thousand times greater than the combined power of all the light radiated by all the stars in the that's the signal we saw in September 2015 a billion years after it happened

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the wave information does not tell us in which direction it came from because each interferometer is a whole sky monitor with very little directional information and having two detectors does give us some directional data for example if the wave came in parallel to the line between the two sites the signals would have registered at the exact same time if the wave was perpendicular to the line we would have seen a time delay of 10 milliseconds because the wave travels 3002 kilometers through the earth at the speed of light what we detected was a wave that came in at an angle that caused delay of 6.9 milliseconds the dotted line represents the distance the wave had to travel for a piece of it to reach the hanford interferometer a little trigonometry gives us the angle of course this angle gives us a circle of possible directions

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interferometers are most sensitive to waves that come in perpendicular to their two arms sensitivity drops off as the incidence Direction departs from the perpendicular the curvature of the earth gives the two LIGO interferometer x' an angle difference of around 27 degrees this creates slight amplitude and phase inconsistencies across the two detectors that enable a narrowing of the probabilities to a smaller portion of the sky here are the most probable directions as seen from Earth the best way to increase the accuracy of our directional findings is to use a third gravitational wave detector to triangulate the source several are under construction or being upgraded to do just that here is a map of current and future gravitational wave observatory x' on the Earth's surface one of the greatest opportunities we have now that we can detect gravitational waves will be the ability to observe events that happen before light was traveling across the cosmos the first 380,000 years after the Big Bang are known as the dark ages as you can imagine there is a lot of guess work that goes into figuring out what happened during that period in the universe's history gravitational waves created within the dark ages may help us untangle that mystery more

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Music free version https://www.youtube.com/playlist?list=PLpH1IDQEoE8RAbxNkOjhpPuqSCo2dF8Ji

In this segment of the “How Fast Is It” video book, we cover Gravitational Waves. We examine just what a ‘ripple in space-time’ is. We’ll cover the wave properties including its metric, speed, and the two transverse polarizations: h plus and h cross. We’ll cover how it expands and contracts objects it encounters. We’ll examine binary star systems and the waves they create. This includes the properties of the binary encoded in the created gravitational wave including frequency, frequency changes (the chirp), chirp mass, gravitational luminosity and luminosity distance. We’ll use stars the size and mass of our Sun to calculate the expected amplitude and wavelength magnitudes. We’ll move from normal stars to neutron stars to stellar mass black holes. Along that progression, we’ll see their signature waveforms and build the gravitational wave sensitivity graph. We’ll cover the Hulse-Taylor pulsar (PSR B1913+16) and how it provided indirect evidence for the existence of gravitational waves. We’ll then cover direct detection: first with strain gauges and then with Michelson Interferometers. We’ll cover the LIGO technology and sensitivity and then examine the GW150914 event. We’ll show how it fit the waveform and magnitudes for merging stellar black holes. We’ll also quantify the magnitude of the radiated energy. Then we’ll cover how we located the sky location for the event. We’ll end with a look at other observatories and the impact detecting gravitational waves will have on cosmology.

Music free version https://www.youtube.com/playlist?list=PLpH1IDQEoE8RAbxNkOjhpPuqSCo2dF8Ji

In this segment of the “How Fast Is It” video book, we cover Gravitational Waves. We examine just what a ‘ripple in space-time’ is. We’ll cover the wave properties including its metric, speed, and the two transverse polarizations: h plus and h cross. We’ll cover how it expands and contracts objects it encounters. We’ll examine binary star systems and the waves they create. This includes the properties of the binary encoded in the created gravitational wave including frequency, frequency changes (the chirp), chirp mass, gravitational luminosity and luminosity distance. We’ll use stars the size and mass of our Sun to calculate the expected amplitude and wavelength magnitudes. We’ll move from normal stars to neutron stars to stellar mass black holes. Along that progression, we’ll see their signature waveforms and build the gravitational wave sensitivity graph. We’ll cover the Hulse-Taylor pulsar (PSR B1913+16) and how it provided indirect evidence for the existence of gravitational waves. We’ll then cover direct detection: first with strain gauges and then with Michelson Interferometers. We’ll cover the LIGO technology and sensitivity and then examine the GW150914 event. We’ll show how it fit the waveform and magnitudes for merging stellar black holes. We’ll also quantify the magnitude of the radiated energy. Then we’ll cover how we located the sky location for the event. We’ll end with a look at other observatories and the impact detecting gravitational waves will have on cosmology.

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