ELEMENTS FOR PHYSICS

By C. NAVAR

CONTENTS 0 OVERVIEW 1 1 GEOTENSORS 11 1 1 LINEAR SPACE 11 1 2 AUTOVECTOR SPACE 18 1 3 ORIENTED AUTOPARALLEL SEGMENTS ON A MANIFOLD 31 1 4 LIE GROUP MANIFOLDS 41 1 5 GEOTENSORS 75 2 TANGENT AUTOPARALLEL MAPPINGS 79 2 1 DECLINATIVE AUTOVECTOR SPACES 81 2 2 DECLINATIVE CONNECTION MANIFOLDS 87 2 3 EXAMPLE MAPPINGS FROM LINEAR SPACES INTO LIE GROUPS 92 2 4 EXAMPLE MAPPINGS BETWEEN LIE GROUPS 100 2 5 COVARIANT DECLINATIVE 102 3 QUANTITIES AND MEASURABLE QUALITIES 105 3 1 ONE DIMENSIONAL QUALITY SPACES 107 3 2 SPACE TIME 118 3 3 VECTORS AND TENSORS 122 4 INTRINSIC PHYSICAL THEORIES 125 4 1 INTRINSIC LAWS IN PHYSICS 125 4 2 EXAMPLE LAW OF HEAT CONDUCTION 126 4 3 EXAMPLE IDEAL ELASTICITY 133 OVERVIEW ONE DIMENSIONAL QUALITY SPACES CONSIDER A ONE DIMENSIONAL SPACE EACH POINT N OF IT REPRESENTING A MUSICAL NOTE THIS LINE HAS TO BE IMAGINED IN NITE IN ITS TWO SENSES WITH THE IN NITELY ACUTE TONES AT ONE END AND THE IN NITELY GRAVE TONES AT THE OTHER END MUSICIANS CAN IMMEDIATELY GIVE THE DISTANCE BETWEEN TWO POINTS OF THE SPACE I E BETWEEN TWO NOTES USING THE OCTAVE AS UNIT TO EXPRESS THIS DISTANCE BY A FORMULA WE MAY CHOOSE TO REPRESENT A NOTE BY ITS FREQUENCY OR BY ITS PERIOD THE DISTANCE BETWEEN TWO NOTES N1 AND N2 IS1 DMUSIC N1 N2 LOG2 2 1 LOG2 2 1 1 THIS DISTANCE IS THE ONLY ONE THAT HAS THE FOLLOWING PROPERTIES ITS EXPRESSION IS IDENTICAL WHEN USING THE POSITIVE QUANTITY 1 OR ITS INVERSE THE POSITIVE QUANTITY 1 IT IS ADDITIVE I E FOR ANY SET OF THREE ORDERED POINTS N1 N2 N3 THE DISTANCE FROM POINT N1 TO POINT N2 PLUS THE DISTANCE FROM POINT N2 TO POINT N3 EQUALS THE DISTANCE FROM POINT N1 TO POINT N3 THIS ONE DIMENSIONAL SPACE OR TO BE MORE PRECISE THIS ONE DIMENSIONAL MANIFOLD IS A SIMPLE EXAMPLE OF A QUALITY SPACE IT IS A METRIC MANIFOLD THE DISTANCE BETWEEN POINTS IS DE NED THE QUANTITIES FREQUENCY AND PERIOD ARE TWO OF THE COORDINATES THAT CAN BE USED ON THE QUALITY SPACE OF THE MUSICAL NOTES TO CHARACTERIZE ITS POINTS IN NITELY MANY MORE COORDINATES ARE OF COURSE POSSIBLE LIKE THE LOGARITHMIC FREQUENCY LOG 0 THE CUBE OF THE FREQUENCY 3 ETC GIVEN THE EXPRESSION FOR THE DISTANCE IN SOME COORDINATE SYSTEM IT IS EASY TO OBTAIN AN EXPRESSION FOR IT USING ANOTHER COORDINATE SYSTEM FOR INSTANCE IT FOLLOWS FROM EQUATION 1 THAT THE DISTANCE BETWEEN TWO MUSICAL NOTES IS IN TERMS OF THE LOGARITHMIC FREQUENCY DMUSIC N1 N2 2 1 THERE ARE MANY QUANTITIES IN PHYSICS THAT SHARE THREE PROPERTIES I THEIR RANGE OF VARIATION IS 0 II THEY ARE AS COMMONLY USED AS THEIR IN 1TO OBTAIN THE DISTANCE IN OCTAVES ONE MUST USE BASE 2 LOGARITHMS 2 OVERVIEW VERSES AND III THEY DISPLAY THE BENFORD E ECT 2 EXAMPLES ARE THE FREQUENCY 1 AND PERIOD 1 PAIR THE TEMPERATURE T 1 AND THER MODYNAMIC PARAMETER 1 T PAIR OR THE RESISTANCE R 1 C AND CONDUCTANCE C 1 R PAIR THESE QUANTITIES TYPICALLY ACCEPT THE EXPRES SION IN FORMULA 1 AS A NATURAL DE NITION OF DISTANCE IN THIS BOOK WE SAY THAT WE HAVE A PAIR OF JE REYS QUANTITIES FOR INSTANCE BEFORE THE NOTION OF TEMPERATURE3 WAS INTRODUCED PHYSI CISTS FOLLOWED ARISTOTLE IN INTRODUCING THE COLD HOT QUALITY SPACE EVEN IF A PARTICULAR COORDINATE OVER THIS ONE DIMENSIONAL MANIFOLD WAS NOT AVAILABLE PHYSICISTS COULD QUITE PRECISELY IDENTIFY MANY OF ITS POINTS THE POINT Q1 CORRESPONDING TO THE MELTING OF SULPHUR THE POINT Q2 CORRESPONDING TO THE BOILING OF WATER ETC AMONG THE MANY COORDINATES TODAY AVAILABLE IN THE COLD HOT SPACE LIKE THE CELSIUS OR THE FAHRENHEIT TEMPERATURES THE PAIR ABSOLUTE TEMPERATURE T 1 AND THERMODYNAMIC PARAMETER 1 T ARE OBVIOUSLY A JE REYS PAIR IN TERMS OF THESE COORDINATES THE NATURAL DISTANCE BETWEEN TWO POINTS OF THE COLD HOT SPACE IS USING NATURAL LOGARITHMS DCOLD HOT Q1 Q2 LOG T2 T1 LOG 2 1 T 2 T 1 2 1 2 WHERE FOR MORE COMPLETENESS THE LOGARITHMIC TEMPERATURE T AND THE LOG ARITHMIC THERMODYNAMIC PARAMETER HAVE ALSO BEEN INTRODUCED AN EX PRESSION USING OTHER COORDINATES IS DEDUCED USING ANY OF THOSE EQUIVALENT EXPRESSIONS FOR INSTANCE USING CELSIUS TEMPERATURES DCOLD HOT Q1 Q2 LOG T2 T0 T1 T0 WHERE T0 273 15 K AT THIS POINT WITHOUT ANY FURTHER ADVANCE IN THE THEORY WE COULD ALREADY ASK A SIMPLE QUESTION IF THE TONE PRODUCED BY A MUSICAL INSTRUMENT DEPENDS ON THE POSITION OF THE INSTRUMENT IN THE COLD HOT SPACE USING ORDINARY LANGUAGE WE WOULD SAY THAT THE FREQUENCY OF THE NOTE DEPENDS ON THE TEMPERATURE BUT WE SHOULD NOT TRY TO BE SPECI C WHAT IS THE SIMPLEST DEPENDENCE THAT WE CAN IMAGINE SURELY A LINEAR DEPENDENCE BUT AS BOTH SPACES THE SPACE OF MUSICAL NOTES AND THE COLD HOT SPACE ARE METRIC THE ONLY INTRINSIC DE NITION OF LINEARITY IS A PROPORTIONALITY BETWEEN THE DISTANCES IN THE TWO SPACES DMUSIC N1 N2 DCOLD HOT Q1 Q2 3 WHERE IS A POSITIVE REAL NUMBER NOTE THAT WE HAVE JUST EXPRESSED A PHYS ICAL LAW WITHOUT BEING SPECI C ABOUT THE MANY POSSIBLE PHYSICAL QUANTITIES 2THE BENFORD E ECT IS AN UNEVEN PROBABILITY DISTRIBUTION FOR THE RST DIGIT IN THE NUMERICAL EXPRESSION OF A QUANTITY WHEN USING A BASE K NUMBER SYSTEM THE PROBABILITY THAT THE RST DIGIT IS N IS PN LOGK N 1 N FOR INSTANCE IN THE USUAL BASE 10 SYSTEM ABOUT 30 OF THE TIME THE RST DIGIT IS ONE WHILE FOR ONLY 5 OF THE TIME IS THE RST DIGIT A NINE SEE DETAILS IN CHAPTER 3 3BEFORE GALILEO THE QUANTITY TEMPERATURE WAS NOT DE NED AROUND 1592 HE INVENTED THE RST THERMOMETER USING AIR OVERVIEW 3 THAT ONE MAY USE IN EACH OF THE TWO QUALITY SPACES CHOOSING FOR INSTANCE TEMPERATURE T IN THE COLD HOT SPACE AND FREQUENCY IN THE SPACE OF MU SICAL NOTES THE EXPRESSION FOR THE LINEAR LAW 3 IS 2 1 T2 T1 4 NOTE THAT THE LINEAR LAW TAKES A FORMALLY LINEAR ASPECT ONLY IF LOGARITHMIC FRE QUENCY OR LOGARITHMIC PERIOD AND LOGARITHMIC TEMPERATURE OR LOGARITHMIC THERMODYNAMIC PARAMETER ARE USED AN EXPRESSION LIKE 2 1 T2 T1 ALTHOUGH FORMALLY LINEAR IS NOT A LINEAR LAW AS FAR AS WE HAVE AGREED ON GIVEN METRICS IN OUR QUALITY SPACES MULTI DIMENSIONAL QUALITY SPACES CONSIDER A HOMOGENEOUS PIECE OF A LINEAR ELASTIC MATERIAL IN ITS UNSTRESSED STATE WHEN A HOMOGENEOUS STRESS IJ IS APPLIED THE BODY EXPERI ENCES A STRAIN IJ THAT IS RELATED TO THE STRESS THROUGH ANY OF THE TWO EQUIVALENT EQUATIONS HOOKE S LAW IJ DIJ K K IJ CIJ K K 5 WHERE D DIJK IS THE COMPLIANCE TENSOR AND C CIJK IS THE STI NESS TENSOR THESE TWO TENSORS ARE POSITIVE DE NITE AND ARE MUTUALLY INVERSE DIJK CK MN CIJK DK MN I M J N AND ONE CAN USE ANY OF THE TWO TO CHARACTERIZE THE ELASTIC MEDIUM IN ELEMENTARY ELASTICITY THEORY ONE ASSUMES THAT THE COMPLIANCE TENSOR HAS THE SYMMETRIES DIJK DJIK DK IJ WITH AN EQUIVALENT SET OF SYMMETRIES FOR THE STI NESS TENSOR AN EASY COMPUTATION SHOWS THAT IN 3D MEDIA ONE IS LEFT WITH 21 DEGREES OF FREEDOM I E 21 QUANTITIES ARE NECESSARY AND SU CIENT TO CHARACTERIZE A LINEAR ELASTIC MEDIUM WE CAN THEN INTRODUCE AN ABSTRACT 21 DIMENSIONAL MANIFOLD E SUCH THAT EACH POINT E OF E CORRESPONDS TO AN ELASTIC MEDIUM AND VICE VERSA THIS IS THE QUALITY SPACE OF ELASTIC MEDIA WHICH SETS OF 21 QUANTITIES CAN WE CHOOSE TO REPRESENT A LINEAR ELASTIC MEDIUM FOR INSTANCE WE CAN CHOOSE 21 INDEPENDENT COMPONENTS OF THE COMPLIANCE TENSOR DIJK OR 21 INDEPENDENT COMPONENTS OF THE STI NESS TENSOR CIJK OR THE SIX EIGENVALUES AND THE 15 PROPER ANGLES OF THE ONE OR THE OTHER EACH OF THE POSSIBLE CHOICES CORRESPONDS TO CHOOSING A COORDINATE SYSTEM OVER E IS THE MANIFOLD E METRIC I E IS THERE A NATURAL DE NITION OF DISTANCE BETWEEN TWO OF ITS POINTS THE REQUIREMENT THAT THE DISTANCE MUST HAVE THE SAME EXPRESSION IN TERMS OF COMPLIANCE D AND IN TERMS OF STI NESS C THAT IT MUST HAVE AN INVARIANCE OF SCALE MULTIPLYING ALL THE COMPLIANCES OR ALL THE STI NESSES BY A GIVEN FACTOR SHOULD NOT ALTER THE DISTANCE AND THAT IT SHOULD DEPEND ONLY ON THE INVARIANT SCALARS OF THE COMPLIANCE OR OF THE