OPTICAL MEASUREMENT TECHNIQUES
CHAPTER 1 THE SEARCH FOR GRAVITATIONAL WAVES CHAPTER 2 GRAVITATIONAL WAVE DETECTOR LIMITATIONS 15 CHAPTER 3 DEVELOPMENT OF A GEOMETRIC SENSOR 31 CHAPTER 4 DEVELOPMENT OF THE INTERFEROMETRIC SENSOR 79 CHAPTER 5 CONCLUSIONS OF THIS STUDY 121 REFERENCES 131 CONTENTS 1 CHAPTER 1 THE SEARCH FOR GRAVITATIONAL WAVES NOTHING HAS SUCH POWER TO BROADEN THE MIND AS THE ABILITY TO INVESTIGATE SYSTEMATICALLY AND TRULY ALL THAT COMES UNDER THY OBSERVATION IN LIFE MARCUS AURELIUS 1 1 INTRODUCTION THROUGHOUT HISTORY HUMANS HAVE BEEN AWARE OF THE FORCE OF GRAVITY FROM THE PROJECTILE WEAPONS THROWN BY PREHISTORIC MAN TO THE MANNED MOON LANDINGS IN THE TWENTIETH CENTURY GRAVITY HAS PLAYED A PIVOTAL PART IN HUMAN ACHIEVEMENTS BUT FOR CENTURIES THE ROLE OF GRAVITY HAS BEEN SILENT AND IT IS ONLY NOW THAT WE ARE AT THE THRESHOLD OF UNCOVERING ITS MYSTERY EARLY EFFORTS MADE AT UNDERSTANDING THE FORCE OF GRAVITY WERE MAINLY QUALITATIVE IN NATURE IT WAS NOT UNTIL LATE IN THE SEVENTEENTH CENTURY THAT A VIABLE THEORY ON GRAVITY WAS PROPOSED IN 1687 SIR ISAAC NEWTON PUBLISHED IN HIS PHILOSOPHI NATURALIS PRINCIPIA MATHEMATICA 1 THE UNIVERSAL LAW OF GRAVITATIONAL ATTRACTION 2 2 1 G R M M F 1 1 WHERE F IS THE MAGNITUDE OF THE GRAVITATIONAL FORCE IN NEWTONS M1 AND M2 ARE THE MASSES IN KILOGRAMS OF TWO SPHERICAL OR POINT MASSES R IS THE DISTANCE BETWEEN THE MASSES IN METRES AND G IS THE NEWTONIAN GRAVITATIONAL CONSTANT THIS INVERSE SQUARE LAW PROVIDED A BREAKTHROUGH FOR THOSE WANTING A MORE QUANTITATIVE UNDERSTANDING OF GRAVITY HOWEVER IT WAS OVER A CENTURY LATER IN 1798 WHEN MEASUREMENTS OF THE DENSITY OF THE EARTH CONDUCTED BY HENRY CAVENDISH 2 ALLOWED A VALUE TO BE DETERMINED FOR THE GRAVITATIONAL CONSTANT G 6 74 10 11 M3 KG 1 S 2 ALTHOUGH NEWTON S UNIVERSAL LAW OF GRAVITATIONAL ATTRACTION APPEARS VALID FOR MANY EARTH BASED OBSERVATIONS IT HAS AN UNDENIABLE FAILING UNFORTUNATELY THIS LAW GIVES 1 NO INSIGHT INTO THE SPEED OF PROPAGATION FOR ANY CHANGE IN THE PARAMETERS OF THE SYSTEM SINCE IT WAS TIME INDEPENDENT AS THIS LAW IS WRITTEN ANY CHANGE OF MASS OR SEPARATION BETWEEN THE MASSES WILL RESULT IN AN INSTANTANEOUS EFFECT ANYWHERE IN THE UNIVERSE HENCE INFORMATION ABOUT THE STATE OF THE SYSTEM WOULD BE REQUIRED TO PROPAGATE FASTER THAN THE SPEED OF LIGHT WHICH WOULD LATER EXPLICITLY BE FORBIDDEN BY EINSTEIN S THEORY OF SPECIAL RELATIVITY 3 1 2 GRAVITATIONAL WAVE THEORY EINSTEIN S GENERAL THEORY OF RELATIVITY 4 PUBLISHED IN 1916 REVOLUTIONISED OUR VIEW OF THE UNIVERSE AND PREDICTED THE EXISTENCE OF GRAVITATIONAL RADIATION SUCH RADIATION IS GENERATED BY THE ACCELERATION OF MASS ENERGY DISTRIBUTIONS AND IS EXPECTED TO BEHAVE LIKE A WAVE LIKE DISTORTION IN SPACE TIME PROPAGATING AT THE SPEED OF LIGHT PROPAGATING GRAVITATIONAL WAVES CHANGE THE CURVATURE OF FOUR DIMENSIONAL SPACE TIME AND CAN CHANGE THE MEASURED SEPARATION BETWEEN FREE MASSES GRAVITATIONAL WAVES ARE OF VERY LOW INTENSITY AND INTERACT WEAKLY WITH MATTER THEY CAN BE BETTER UNDERSTOOD BY ANALOGY TO ELECTRO MAGNETIC E M WAVES THE MOTION OF A PARTICLE CARRYING AN ELECTRIC CHARGE CAN GENERATE E M RADIATION JUST LIKE THE MOTION OF A MASS CAN PRODUCE A GRAVITATIONAL WAVE SINCE BOTH WAVES ARE TRANSVERSE AND PROPAGATE AT THE SPEED OF LIGHT HOWEVER THE FUNDAMENTAL DIFFERENCE IS THAT BOTH POSITIVE AND NEGATIVE CHARGES EXIST FOR E M INTERACTIONS WHEREAS THE MASS OF A PARTICLE CAN ONLY HAVE ONE SIGN THIS LEADS TO GRAVITATIONAL RADIATION BEING QUADRUPOLE IN NATURE WITH MONOPOLE AND DIPOLE RADIATION BEING FORBIDDEN 5 EINSTEIN PROPOSED THE FIELD EQUATION SHOWING THE INTERACTION BETWEEN THE CURVATURE OF SPACE TIME AND THE MASS DISTRIBUTION T G 1 2 WHERE G IS DEFINED AS THE EINSTEIN CURVATURE TENSOR T IS THE STRESS ENERGY TENSOR REPRESENTING THE MASS ENERGY DISTRIBUTION AND DESCRIBES THE EINSTEIN CONSTANT OF GRAVITATION WHICH IS DEFINED AS 4 G 8 C 1 3 2 WHERE C IS THE SPEED OF LIGHT IN A VACUUM IT IS NECESSARY FOR THE VALUE OF CONSTANT TO ENABLE THE EINSTEIN APPROACH TO BE CONSISTENT WITH THE NEWTONIAN APPROACH IN THE WEAK AND SLOWLY VARYING GRAVITATIONAL FIELD REGIME THE LARGE COUPLING CONSTANT IMPLIES SPACE TIME MUST BE A VERY STIFF AND ELASTIC MEDIUM FIGURE 1 1 LOCATIONS OF FREE TEST MASSES FOR BOTH POLARISATIONS AND OF A PROPAGATING GRAVITATIONAL WAVE SOURCE 6 THE EFFECTS OF A GRAVITATIONAL WAVE CANNOT BE OBSERVED FOR ISOLATED BODIES BUT ONLY BY OBSERVING THE CHANGE IN PROPER DISTANCE MEASURED BY A LASER BETWEEN PAIRS OF MASSES THEREFORE TO HELP VISUALISE THE EFFECT OF A PROPAGATING GRAVITATIONAL WAVE CONSIDER A RING OF FREE TEST MASSES AS SHOWN IN FIGURE 1 1 AS THE GRAVITATIONAL WAVE PROPAGATES NORMAL TO THE PAGE IT WILL PERTURB THE DISTANCE BETWEEN THESE TEST MASSES ASSUMING THAT THIS GRAVITATIONAL WAVE IS OF THE POLARISATION THEN DURING THE FIRST HALF OF ITS PERIOD TEST MASSES WILL BE STRETCHED APART ALONG THE HORIZONTAL AXIS AND SQUEEZED TOGETHER ALONG THE VERTICAL AXIS DURING THE SECOND HALF OF THE GRAVITATIONAL WAVE S PERIOD THE STRETCHING AND SQUEEZING AXES WILL BE REVERSED IN FIGURE 1 1 THE EFFECT UPON THE TEST MASSES FOR THE POLARISATION OF GRAVITATIONAL WAVE IS SHOWN IN THIS CASE THE AXES OF THE PERTURBATIONS ARE ROTATED THROUGH 45O THE MAGNITUDE OF A GRAVITATIONAL WAVE IS CONVEYED BY THE DIMENSIONLESS STRAIN AMPLITUDE L L T H 2 1 4 3 OR BY ITS SPECTRAL COMPONENT V H WITH UNITS OF HZ 1 2 WHERE L IS THE UNPERTURBED LENGTH AND L IS THE MEASURED CHANGE IN LENGTH FOR THE PURPOSE OF FIGURE 1 1 THE STRAIN AMPLITUDE IS EXAGGERATED TO DEMONSTRATE MORE CLEARLY THE INFLUENCE OF THE PROPAGATING GRAVITATIONAL WAVE ON THE FREE TEST MASSES HOWEVER IN REALITY GRAVITATIONAL WAVES GENERATED BY TYPICAL SOURCES ARE MOST LIKELY ONLY HAVE STRAIN AMPLITUDES OF 10 21 AND ARE ONLY EXPECTED TO OCCUR WITH RATES OF UP TO A COUPLE OF EVENTS PER YEAR 7 1 3 SOURCES OF GRAVITATIONAL WAVES SOURCES OF GRAVITATIONAL WAVES AT LEAST THE ONES THAT ARE MOST LIKELY TO BE DETECTED ARE ASTROPHYSICAL IN NATURE THESE ASTROPHYSICAL SOURCES MAY RADIATE AWAY A SIGNIFICANT PROPORTION OF THEIR ENERGY IN THE FORM OF GRAVITATIONAL WAVES IT IS ALSO EXPECTED THAT SUCH EMISSIONS WILL OCCUR OVER A WIDE BAND OF FREQUENCIES 10 4 HZ TO 1010 HZ WHAT SHOULD WE EXPECT TO OBSERVE THE FOLLOWING SECTIONS PROVIDE SOME INSIGHT INTO THE VARIOUS MECHANISMS BY WHICH ASTROPHYSICAL SOURCES ARE PREDICTED TO EMIT GRAVITATIONAL WAVES IT SHOULD BE NOTED THAT SOURCES ALREADY MODELLED MAY BE JUST A FRACTION OF WHAT WILL BE DISCOVERED WHEN NEW OBSERVATORIES COME ONLINE AND DETECTION BECOMES COMMONPLACE 1 3 1 CHIRP SIGNALS CHIRP SIGNALS ARE ASSOCIATED WITH THE COALESCENCE IN BINARY SYSTEMS WHICH COULD CONTAIN A COMBINATION OF NEUTRON STARS AND OR BLACK HOLES IN SUCH SYSTEMS ANGULAR MOMENTUM IS LOST DUE TO THE EMISSION OF GRAVITATIONAL RADIATION THIS WILL EVENTUALLY LEAD TO THE TWO OBJECTS IN SPIRALLING TOWARDS EACH OTHER AS THIS OCCURS THE FREQUENCY AND AMPLITUDE OF THE GRAVITATIONAL WAVE EMISSIONS WILL INCREASE UNTIL FINALLY THE BODIES MERGE AND COALESCE IF THE BINARY SYSTEM WERE TO CONTAIN TWO BLACK HOLES 10 MA WHERE MA IS THE SOLAR MASS MA 1 99 1030 KG THEN DUE TO THE GREATER MASS OF THE SYSTEM THE COALESCENCE OF THESE BODIES WOULD LEAD TO THE EMISSION OF STRONGER GRAVITATIONAL RADIATION 4