 systems (it has a physical constancy) and is usually expressed in terms of tensor fields (O'Hanian and Ruffini, 1994). Tensors are mathematical objects which describe relationships; this makes them inherently generally covariant and therefore useful in representing general covariance. In Newtonian physics, the universe is assumed to be totally Euclidean, and we use the 3D Euclidean metric to define all of our distances. When time is taken as a fourth dimension, the Galilean transformations preserve the 3D Euclidean distance and time separately in all Euclidean inertial frames. Einstein realized that 3D Euclidean geometry was insufficient to describe electromagnetic effects and the constancy (general covariance) of c; Einstein consequently developed his special theory of relativity in 1905 to explain these effects (Einstein, 2001). The mathematical structure of Minkowski space-time was an immediate consequence of the postulates of special relativity (Landau and Lifshitz, 1960; Petkov, 2010). In Minkowski space-time, space and time are interwoven. The shortest distance between two points (events) in Minkowski space-time is given by the invariant space-time interval between the events, which takes into account not only their spatial separation, but their temporal separation as well (Petkov, 2010). Minkowski space-time is a combination of 3-dimensional Euclidean space and time into a 4-dimensional manifold (coordinate system) where the space-time interval between any two events is independent of the inertial frame of reference in which it was recorded, i.e., space-time intervals are generally covariant (constant) for all Minkowski inertial frames. In Minkowski space-time, the invariant space-time interval replaces the invariant regular Euclidean distance of Euclidean space. The Euclidean metric (Pythagorean Theorem) defines the invariant regular Euclidean distance in Euclidean inertial frames. The Minkowski metric defines the invariant space-time interval in Minkowski inertial frames. The Minkowski metric is defined as to yield the invariant space-time interval between two events when given their coordinate difference vectors (Petkov, 2010; Schutz, 2004): (space-time interval) "s2" is equal to the difference between the space coordinates "∆r2" of two events minus the difference between the time coordinates "c2∆t2" of the two events (space-time interval) s2=∆r2-c2∆t2 or s2=∆r2-∆t2, if c=1 This is the Minkowski metric;
systems (it has a physical constancy) and is usually expressed in term of tensor field (O'Hanian and Ruffini, 1994). Tensors are mathematical objects which trace relationships; this makes them inherently broadly covariant and consequently useful in representing general covariance. In Newtonian physics, the universe is assume to be totally Euclidean, and we use the 3D Euclidean metric to define all of our distance. When time is taken as a fourth dimension, the Galilean transformation preserve the 3D Euclidean distance and time separately in all euclidian inertial frames. Einstein gain that 3D Euclidean geometry was insufficient to describe electromagnetic effects and the constancy (cosmopolitan covariance) of c; Einstein consequently developed his special theory of relativity in 1905 to explain these effects (Einstein, 2001). The numerical social organization of Minkowski outer space - time was an immediate consequence of the postulates of special relativity (Landau and Lifshitz, 1960; Petkov, 2010). In Minkowski space - time, quad and clock time are interwoven. The shortest distance between two point (event) in Minkowski space - time is given by the changeless space - meter interval between the events, which accept into account not only their spatial separation, but their temporal separation as well (Petkov, 2010). Minkowski space - time is a combination of 3 - dimensional Euclidean space and time into a 4 - dimensional manifold (coordinate arrangement) where the space - clock time interval between any two events is independent of the inertial frame of reference in which it was commemorate, i.e., space - time intervals are generally covariant (constant) for all Minkowski inertial frames. In Minkowski space - time, the changeless space - time interval replaces the invariant even Euclidean distance of euclidian space. The Euclidean metric (Pythagorean Theorem) defines the changeless regular Euclidean distance in Euclidean inertial skeletal system. The Minkowski metric defines the invariant space - time interval in Minkowski inertial frames. The Minkowski metric is defined as to yield the invariant outer space - time interval between two consequence when given their coordinate difference vector (Petkov, 2010; Schutz, 2004 ): (space - time interval) " s2 " is adequate to the difference between the space coordinates " ∆r2 " of two events minus the difference between the time coordinates " c2∆t2 " of the two events (quad - clock time interval) s2=∆r2 - c2∆t2 or s2=∆r2-∆t2, if c=1 This is the Minkowski metric;
 sydtems (it has a physical gonstancy) and is usuallb expreased in germs of tensor fields (O'Hanien abd Rudfini, 1994). Tensors are matfematical objects whirh describe relavjonships; this lakev them inherentky generalny covariant atd tkerefore useful in representing genewal covsrlance. In Newtogian [hysjbs, the universe is assumed to bs totalny Euclidean, snd we use the 3D Euclidean metgic to define all lf our distqncef. When time ir taken as a fourth dijension, the Galilean transformagions preserve jkw 3D Guclidean diwtancv and time seisratelf in alk Euclidean incrtian feames. Einstein realizxd that 3D Euclidean deometry faa insufficient to dwscrite enectfimaeneuic erfects anv the constzncy (generao covariance) of c; Eonfnrin consequehtly dqvqloped his special theory of relativity in 1905 to explain these effexts (Einstein, 2001). The matjematical structure of Minkowski space-time was an immediata conadqutngc of rhf postulates of special relativity (Landau and Migsmitz, 1960; Petkov, 2010). In Minkowski xpwcr-jime, space and time cde interwoven. The shlrtest qistabce betwetn twp points (events) in Minkowsku space-time ps guven by the invaricnt space-timz intetval brtween the events, which takss into acclunt not kvly their spatiau svpardtion, buu their temporal sqparation as cell (Petyov, 2010). Minkorski space-hime lv a combination of 3-dimeusiondl Euclidewn space and time into a 4-dimensmpnal manifolc (woogdinate svstem) ehere the spase-time intervak betwezn any two eventa is invependent of the inertial vrame of refxrence in whixh ir was rdzorded, i.e., spacr-time intvrrals are tenerally covarianb (conrfant) for all Miuyiwski inertial graoes. Ij Kignowski space-dime, thd invafiant skcce-bimd inyerval replaces the hnvadiant regular Euclodcan distabce of Etclidean spacr. The Euclidean meuric (Pbthagocean Tneotem) defines the invariant regulzr Euclidfan distance in Eucoidean inertnal frames. The Minkowski metric defines vhe invariant space-timg interval in Minkowsll inertial fcames. Ehe Minkofski metric is definwd as to yield tht invariant space-time ihtervan betaeen two events when given their coordinate difference vectors (Petkov, 2010; Scyutz, 2004): (space-time infervsl) "s2" ps zqucl to tre dmfherence between tne space coordinates "∆r2" of two etents minuv che difference between the tike coordinates "z2∆t2" of the two events (spade-time imterval) s2=∆r2-c2∆t2 or s2=∆r2-∆t2, if c=1 This ix the Minkowski metric;
systems (it has a physical constancy) and expressed terms of fields (O'Hanian and objects describe relationships; this them inherently generally and therefore useful in representing general In Newtonian physics, the universe is assumed to be totally Euclidean, and we the 3D Euclidean metric to define all of our distances. When time is as fourth the transformations preserve the 3D Euclidean distance and time separately in all Euclidean inertial frames. Einstein realized 3D Euclidean geometry was insufficient to describe electromagnetic and the constancy (general of c; Einstein consequently developed special of relativity 1905 explain effects (Einstein, 2001). mathematical structure of Minkowski space-time was an immediate consequence of the postulates of special relativity (Landau and 1960; Petkov, Minkowski space-time, and are The shortest distance points (events) in Minkowski space-time is invariant space-time interval between the events, which takes account not their spatial separation, but their temporal as well (Petkov, 2010). Minkowski space-time is a of 3-dimensional Euclidean space and time into a 4-dimensional manifold (coordinate system) where the space-time any two events is of the inertial of in it recorded, i.e., intervals are generally covariant (constant) for all Minkowski inertial frames. In space-time, the invariant space-time interval replaces the invariant regular Euclidean Euclidean The Euclidean metric Theorem) defines the invariant Euclidean in Euclidean inertial frames. metric the in inertial The Minkowski metric is as to yield the invariant interval between two events vectors (Petkov, 2010; Schutz, 2004): (space-time interval) "s2" equal to the difference between the space "∆r2" of two events minus the difference between the time coordinates "c2∆t2" the two interval) s2=∆r2-c2∆t2 or s2=∆r2-∆t2, if c=1 This is Minkowski metric;
 systems (it has a physical consTancy) and is UsualLy eXprEsSed iN terMs of tensor fielDS (O'HaNian and Ruffini, 1994). Tensors aRe matHeMAticAL oBjectS which dEScRIBe rElAtIonShIPs; This mAkeS them inHerently geNerAlLy covariant aND tHerefore usEfuL in representIng GeneraL cOvaRIance. in NEwtonIan phySIcs, the Universe iS aSSumed tO Be totalLY euClidEan, and we use the 3D EuCLiDEan metric to defIne all Of OUr DIStaNceS. When time iS tAken aS A fourth DImENSIon, THe Galilean traNsformationS PreServe tHe 3d EuCLidean DistaNcE And Time separatEly iN all EucliDean inERtial frAMes. EinsTein reAliZed That 3d euClIdeAn GEomETrY waS InsUfficienT tO dEscriBe elECTROmagNetIc efFects And the constanCy (gEnerAL coVariaNce) of C; EinStEin coNsequeNtly dEvEloped his speciaL theOry of relaTivItY in 1905 To ExplaIN these EffEctS (EinsteIn, 2001). The maTHemAtICAL sTructure of MinkowskI sPACe-Time was aN immedIAtE cONsequencE oF thE posTULates Of spECiAl relatiVity (LaNDaU aNd LifshItZ, 1960; PetkoV, 2010). IN MiNkoWski sPAce-tIme, spaCe and timE are iNTerwoven. The shoRTest distance bETwEEN tWO poiNts (Events) in MinKowsKI spaCe-tiME iS giVEn by tHe invArIAnT Space-time interval beTwEen the EventS, which takes inTo account nOT ONly their SpatIAl SEparation, but thEir teMporal sepaRAtion as wEll (PeTkov, 2010). MinkOwski spacE-TIme is a coMbiNatIon Of 3-dIMEnSional EuclideAN SpacE aNd time iNto A 4-dimensIonAl mAniFolD (cOordinate System) whErE tHe SpAce-Time iNTerval beTwEen AnY twO evenTS is indEpendEnt oF tHe INerTial fraME oF REferEnCe In whIch It Was reCordED, i.e., Space-tiMe intervaLs aRE genErAlLy covarIant (constant) fOr All MinkowsKi IneRtial fRAMes. In MinKowski space-time, the invarIAnt spacE-tiMe intErvaL replaces The InvariAnt REgular euclidEan diStAncE OF EuclIDEaN spAcE. The EuclidEAN meTric (PYtHagoRean TheOrem) defines the invaRIanT regular EucliDeaN disTANcE in eUcLIdeAn INerTIAl frames. The MinkOwski metriC dEFiNes the invaRIanT sPace-timE intervAl in MINkowski Inertial fRames. The MInKowsKI MetRic is definEd as to yiEld the invARiant SPaCe-timE inTerval BeTweEn two Events WHen Given Their cOoRdinatE diffErEnce vectOrs (Petkov, 2010; Schutz, 2004): (space-timE interVal) "s2" iS eqUal to the dIffERenCe between The sPace coordiNatEs "∆r2" Of two EveNTs minUs thE DiFfeREnce bEtweEN the time cOOrDinATEs "C2∆t2" of the two eVENTs (sPace-tIme INtervaL) s2=∆r2-c2∆T2 or s2=∆r2-∆t2, if c=1 This is thE minkowski metriC;
 systems (it has a physica l constanc y) an d i s u su ally  exp ressed in term s  oftensor fields (O'Hania n and  R u ffin i ,1994) . Tenso r sa r e m at he mat ic a lobjec tswhich d escribe re lat io nships; this ma kes them i nhe rently gener all y cova ri ant and t her efore  usefu l  in re presentin gg eneral covaria n c e.  InNewtonian physics ,  t h e universe isassume dt ob e  to tal ly Euclide an , and we uset he 3 D  Eu c lidean metric  to definea llof our  d ist a nces.Whenti m e i s taken asa fo urth dime nsion, the Gal i lean tr ansfor mat ion s pr e se rv e t he 3DE uc lid e andistance  a nd  time  sep a r a t elyinallEucli dean inertial  fr ames .  Ei nstei n rea lize dthat3D Euc lidea ngeometry was in suff icient to  de sc rib eelect r omagne tic  ef fects a nd thec ons ta n c y  ( general covariance )o f  c ; Einste in con s eq ue n tly deve lo ped  his s pecia l th e or y of rel ativit y  i n1905 to  e xplain  t hes e e ffect s  (Ei nstein , 2001).  Them athematical st r ucture of Min k ow s k is pace -ti me was an i mmed i atecons e qu enc e  of t he po st u la t es of special relat iv ity (L andau  and Lifshitz , 1960; Pe t k o v, 2010) . In Mi n kowski space-t ime,space andt ime areinter woven. T he shorte s t  distanc e b etw een  tw o po ints (events) i n Mi nk owski s pac e-timeisgiv enbyth e invaria nt space -t im ein ter val b e tween th eeve nt s,which takesintoacco un tn otonly th e ir s pati al  s epar ati on , but  the i r t emporal  separati ona s we ll  ( Petkov,  2010). Minko ws ki space-t im e i s a co m b inationof 3-dimensional Euclid e an spac e a nd ti me i nto a 4-d ime nsiona l m a nifold  (coor dinat esys t e m) wh e r ethe  s pace-timei n ter val b et ween  any tw o events is indepe n den t of the iner tia l fr a m eofr ef e ren ce inw h ich it was reco rded, i.e. ,s pa ce-time in t erv al s are g enerall y cov a riant ( constant)  for allMi nkow s k i i nertial fr ames. In  Minkowsk i  spac e -t ime,the  invar ia ntspace -timei nte rvalreplac es  the i nvari an t regula r Euclidean distance of  Eucli deanspa ce. The E ucl i dea n metric(Pyt hagorean T heo rem ) def ine s  theinva r ia ntr egula r Eu c lidean di s ta nce i nEuclidean i n e r tia l fra mes .  The M inko wski metric defin e s the invarian t sp a c e-t ime inte rv al in Minkowsk i i ne r t ial fram es . The Minko wski met ri c  is d efined  as to  yieldt h ei nvaria nt s pac e-time in ter va l  betwee ntw o  event s wh en  given  their coor d i nate differencevecto r s  (Pet k ov,  2010 ;Schutz, 2004 ): (space- time interv al) "s 2" i s equ al to t he  diffe ren ce  between t h e space c oordi nates " ∆r 2" o f t wo eve ntsm i nus t he d if fer ence betw e e nt he  t i mecoor dinat es  "c2 ∆t2" of t h e two ev ent s  (space -t ime i nterva l )s 2 =∆r2-c2∆t2  or  s2=∆ r 2 -∆t2, if c = 1 Th i si s the  Minko wski m etric;
 systems_(it has_a physical constancy) and_is usually_expressed_in terms_of_tensor fields (O'Hanian_and Ruffini, 1994)._Tensors are mathematical objects_which describe relationships;_this_makes them inherently generally covariant and therefore useful in representing general covariance. In Newtonian_physics,_the universe_is_assumed_to be totally Euclidean, and_we use the 3D Euclidean_metric to_define all of our distances. When time is_taken_as a fourth_dimension, the Galilean transformations preserve the 3D Euclidean distance_and time separately in all Euclidean_inertial frames. Einstein_realized_that_3D Euclidean geometry was_insufficient to describe electromagnetic effects and_the constancy (general covariance) of c;_Einstein consequently developed his special theory of_relativity in 1905 to explain these_effects (Einstein, 2001). The mathematical_structure of_Minkowski space-time was an immediate_consequence of the_postulates of_special relativity (Landau_and Lifshitz, 1960; Petkov, 2010). In_Minkowski space-time, space_and time are interwoven. The shortest_distance_between two points_(events)_in_Minkowski space-time_is given by_the_invariant space-time_interval_between the events, which takes into_account_not only their spatial separation, but their_temporal separation as well_(Petkov,_2010). Minkowski space-time is_a combination of 3-dimensional Euclidean_space and time into a 4-dimensional_manifold (coordinate_system) where_the space-time interval between any two events is independent of the_inertial frame of reference in which_it was recorded, i.e.,_space-time intervals_are_generally covariant (constant)_for_all Minkowski_inertial frames. In Minkowski space-time, the invariant_space-time interval_replaces the invariant regular Euclidean distance_of Euclidean space. The_Euclidean_metric (Pythagorean Theorem) defines the invariant_regular Euclidean distance in Euclidean inertial_frames. The Minkowski metric defines_the_invariant_space-time interval in Minkowski inertial_frames. The Minkowski metric is defined_as to yield_the invariant space-time interval between two events_when_given their coordinate difference vectors (Petkov,_2010;_Schutz, 2004): (space-time interval) "s2" is_equal_to_the difference between the space_coordinates "∆r2" of two events minus_the difference between the time coordinates "c2∆t2" of the_two events (space-time_interval) s2=∆r2-c2∆t2 or s2=∆r2-∆t2, if_c=1_This_is the Minkowski metric;