arXiv:1802.05290v1 [gr-qc] 14 Feb 2018 Email: codn oFec sepg 6 [ 66, page (see French to According Introduction 1 . of structure causal the of ori- part spacetime as the of entation concept a of importance demon- the and relativity, strate of invari- theory taken Lorentz general the of be under principle full ance not the the examine can We under transformations laws granted. 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Galileo of and the by also satisfied equivalence, ainppr nterltvt hoy ntespecial the In [ theory. paper relativity relativity the on papers dation of laws physical Nature. determining for guide important and an potential, is gravitational arti- the this on heavily in depends relativity cle, general of examined setting be the in will carefully In- itself, Lorentz formulation of whose Principle variance, The is relativity, of principle nEnti [ Einstein in mov- a by or body. stationary velocitying a by determined emitted be the ray the with whether co-ordinates of tem of other mo- the translatory tion. uniform or in one coordinates the of to systems two referred these be whether state affected, of not changes are change undergo tems relativity. of theory his vance fc-riae,ta s r ovratwt respect with co-variant are is, systems ex- that all be co-ordinates, for to good of are hold which nature equations of by laws pressed general [ “The 117, that stated page the In (see paper systems”. “coordinate same “systems of phase instead the of reference” kind used of any Einstein in nature here reference Note a of such .” systems of to be apply must they physics that of laws “The he and wrote relativity, of principle the re-examined Einstein (page chosen.” [ coordinates 39, the a of in independent -pulses of form of motion and of particles laws material “the undisturbed demands equivalence of ciple esalbgnwt htEnti adi i foun- his in said Einstein what with begin shall We h rtpsuaecle h rnil frelativity of principle the called postulate first The sys- “stationary” the in moves light of ray Any 2. sys- physical of states the which by laws The 1. 8 22 ) twsaraycert digo htthe that Eddington to clear already was It ]). ) isenfruae w otltst ad- to postulates two formulated Einstein ]), ∗ 8 ) hrfr,wt h salsmn of establishment the with Therefore, ]). 9 ,i oeotncle h rnil of principle the called often more is ], 9 ,pbihdi 95(e ..page e.g. (see 1905 in published ], 22 10 ) oee,Einstein however, ]), ngnrlrelativity, general on ] 8 ,teprin- the ], c , to any substitutions whatever (generally co-variant).” 2 The This version of principle of relativity is too narrow which deprived the important component in the pre- We are in a to give precise meanings for the vious formulation Einstein stated. It is not known why context of the principle of relativity, to motivate our in- Einstein retreated from the principle of relativity, to a terpretation of Lorentz invariance in the general theory narrow version of the covariance principle. of relativity. In the only relativity textbook [2] endorsed by Ein- stein, Bergmann wrote (see page 154) that “The princi- 1. The and the principle of ple of equivalence was ostensibly a fundamental prop- constancy of light . The mathematical models erty of the gravitational forces”, and “the equivalence of spacetime in special and general theories of relativ- of gravitational and interial fields (which is a con- ity (see [9, 10]) were established based on two postu- sequence of the equality of gravitational and inertial lates, the first is the equivalence principle which does masses) gave the principle of equivalence its name”. not involve any physics (see e.g. Norton [24] for a re- About the principle of , Bergmann view about the meaning of this postulate), the second stated (on page 159, [2]) that “The hypothesis is that is the principle of constancy of light velocity, which the geometry of physical space is represented best by implicitly requires that physical processes propagate a formalism which is covariant with respect to general with speed less than or equal to the velocity c of light. coordinate transformations, and that a restriction to a Together, in the special , the two less general group of transformations would not sim- postulates lead to the mathematical model of space- plify that formalism, is called the principle of general time, the . If the motion of a particle covariance”. Therefore, the equivalence principle is is described by space coordinates x,y and z, parame- used to construct mathematical models for spacetime terized by a time variable t, then the postulate of con- only, while the is a mathemat- stancy of light speed demands thatx ˙2 + y˙2 + z˙2 ≤ c2, ical requirement for constructing fields. which can be written in terms of differentials, i.e. 2 2 2 2 2 Let us look at what Pauli said about the principle c dt − dx − dy − dz ≥ 0. The last inequality im- 4 of equivalence. On page 145, [25], Pauli formulated mediately leads to the Minkowski metric on R , so the general version of the equivalence principle as that concepts of space and time are united in a natural way “For every infinitely small world region (i.e. a world in this model. In the general theory of relativity, the region which is so small that the space- and time- two postulates are applied to develop general contin- variation of can be neglected in it) there always uum space-time models being of dimension 2 µ ν exists a – in which has four, together with a Lorentz metric ds = gµν dx dx no influence either on the motion of particles or any with signature (1,−1,−1,−1). The core element of other physical processes.” What Pauli stated, wrote the general theory of relativity is however the postu- just after the creation of Einstein’s general theory of late that the gravitation is described by the curvature relativity, is a very precise statement of the equiva- field associated with the gravitational potential lence principle – the gravitational potential can not be (gµν ). detected locally. Taking a Lorentz (M,gµν ) as a model of In [28], Weyl formulated the principle of relativ- spacetime, the principle of equivalence is no longer ity of Galilei and Newton as the following (page 154, a postulate rather than a mathematical statement: for [28]): “it is impossible to single out from the sys- every x ∈ M, there is a local coordinate sys- tems of reference that are equivalent for mechanics and tem about x (i.e. there is a system of reference near of which each two are correlated by the formulae of x), such that the gravitational potential (gµν ) is di- transformation III special systems without specifying agonalized to coincide with the Minkowski metric individual objects”, here a transformation III means a diag(1,−1,−1,−1) (but this may be true only at x). linear transformation of following type Therefore, the gravitational potential field can not be detected locally. The gravitational effect however can ′ ′ ′ not be wiped away, this is because for whatever the x1 = a11x1 + a12x2 + γ1t + α1 ′ ′ ′ local coordinate system we choose to describe physi- x2 = a21x1 + a22x2 + γ2t + α2 ′ cal processes, one can not write off the curvature even t = t + a locally. where ai j, γk, αk and a are constant. Thus Weyl in- 2. The principle of covariance. Let us exam- cluded the principle of Lorentz invariance (though in a ine the covariance principle for Einstein’s spacetime restricted sense) as a component of relativity principle. (M,gµν ), which is a principle emphasized in litera-

2 ture, it is however a basic requirement for describing der these reflections1, although all the exact laws of physical processes in a way mathematically meaning- nature so far known do have this invariance.” In 1956 ful. Upon acceptance of the model for event space as Lee and Yang [21] suggested explicitly that parity con- a four dimensional manifold M, even without spec- servation might be violated in weak interactions by an- ifying the gravitational potential (gµν ), the principle alyzing the data which demonstrated meson states τ of covariance, under the current setting of theoretical and θ with almost identical mass but decayed to fi- physics, demands that fields describing physical pro- nal states of opposite parity, which was shortly veri- cesses, either classical or quantized, must be sections fied experimentally by Wu [30]. On the other hand, all of fiber bundles over the four dimensional manifold M known experimental data in high physics sup- of spacetime. If derivatives of fields (to formulate dy- port the Lorentz invariance principle (in the context namic equations of fields) are required, then only co- of ): laws of Nature are un- variant derivatives (including exterior derivatives) are der the sub-group L0 of all proper and orthochronous allowed, so that additional source of fields of differ- Lorentz transformations and the translations. ential one forms, i.e. vector Boson fields, must be in- The strong form of the Lorentz invariance perhaps troduced in to the physical processes. Therefore, the is good for dynamics, however it is violated at the covariance principle itself can not be applied to deter- sub-atomic level, the principle of Lorentz invariance, mine physics laws. which should be applicable for high energy physics with the general theory of relativity, has not been ex- 3. The principle of Lorentz invariance. The im- amined in literature. In the past, according to Dirac portant component of the relativity principle is the [6] for example, there was no need to consider grav- principle of Lorentz invariance for laws of Nature. itational effect at the sub-atomic level, and therefore In fact, the geometric dynamics, including Einstein’s the principle of Lorentz invariance was not in de- field equation, has been developed under the help of a mand. However, the successful detection of gravita- strong version of the Lorentz invariance in both special tional wave radiations originated from a pair of merg- and general theories of relativity, which can be stated ing black holes (see [4, 1]) and the more recent de- that geometrical dynamic equations describing macro- tection (see [18, 29, 20, 7, 13]) from a neutron star scopic are invariant under diffeomorphisms merger confirmed one of predictions made by Ein- stein [11] derived from the general theory of relativ- which preserve the gravitational field (gµν ). Einstein showed this strong form of Lorentz invariance of the ity [10, 12], and understanding the mechanics of grav- Maxwell equations, with which the geometric dynam- itational waves requires high energy physics and the ics were developed later by Planck and others. theory of gravitation, in which the Lorentz invariance plays an important role. The aim of this note is to for- Let us examine the principle of Lorentz invariance mulate a version of the principle of Lorentz invariance for the special theory of relativity. Recall that the within the general theory of relativity. A correct for- isometry group of the Minkowski spacetime can be mulation of the Lorentz invariance is crucial for deter- identified with the Poincaré group P. A homor- mining laws of Nature. phism Λ of the Minkowski spacetime is an isome- try if it also preserves the Minkowski metric (ηµν )= (1,−1,−1,−1). Under the standard coordinate sys- 3 Spacetime orientation as causal tem (xµ ) in R4, an isometry must take the form µ µ ν µ µ structure (Λx) = Λ ν x + a , where (Λ ν ) is a constant such that Λ σ η Λ ρ = η (such a transformation is µ σρ ν µν Let us now consider a spacetime (M,gµν ). In order to µ called a ) and a = (a ) is a con- develop a theory of Lorentz invariance with the gen- R4 stant vector in . Therefore, the isometry group in the eral relativity, a careful study of causal structure of special relativity is identified with the Poincaré group spacetime seems necessary. In history, the P L R4 L = ⊕ , where = SO(1,3) is the Lorentz as an important component of the spacetime model group. For the geometric dynamics, it has been pos- emerged, surprisingly, firstly not from the investiga- tulated that the laws of geometric dynamics in special tion of cosmology and gravitation, but from the high P relativity are invariant under the Poincaré group . energy physics. The current knowledge supports the However, at the level of high energy physics, it was postulate that it is the causality structure of the space- recognized that the invariance of laws under the full time which is responsible for the concepts of anti- P isometry group can not be taken as granted. As matters and violations of various discrete symmetri- early as in 1949, Dirac [5] wrote that “I do not believe there is any need for physical laws to be invariant un- 1The space and time inversions.

3 ces. The causality structure became prominent, largely O(M)/L0 to M, and O(M)/L0 is a principal fiber due to the discovery of singularities of the spacetime in bundle over M with its structure group L and typi- 1960’s in the seminal work by Hawking and Penrose cal fiber L /L0 which is a discrete group of four ele- (see e.g. [26, 15, 16, 17]). Here we only need one ele- ments. ment of causal structures which is related to spacetime It is said that a spacetime orientation exists on the orientation. spacetime M, if the structure group L of O(M) can The best way to describe spacetime orientation is be reduced to the Lorentz L0, in the sense to use the language of principal fiber bundles (see e.g. that there is a principal subbundle L(M) of O(M) over Hirzebruch [19]). Let P(M) denote the fiber bundle M with both its structure group and typical fiber being of all frames (also called tetrads) (x,(ek)), where (ek) L0. According to Hirzebruch ([19] Theorem 3.4.5 on is a linear of the tangent space Tx(M) at point page 45), the structure group L of O(M) can be re- µ x ∈ M. Suppose (x ) is a coordinate system near x, duced to L0, if and only if there is a global (smooth) l ∂ L then a linear basis can be expressed as ek = x k ∂xl in section of the principal fiber bundle O(M)/ 0 over terms of partial coordinate derivatives, and therefore, M. A (smooth) section of the principal fiber bundle µ l L it leads to a coordinate system (x ,x k) for the frame of O(M)/ 0 is called a spacetime orientation of the bundle. P(M) is a principle fiber bundle with its struc- spacetime (M,gµν ). ture group the general linear group GL(4), so that, if i a = (a j) is an invertible real 4 × 4 matrix, then its j 4 Principle of Lorentz invariance (effective) right action sends a frame (ek) to (a ke j), which leads to the transformation of the correspond- ing coordinates To be able to implement the principle of Lorentz in- variance in the general theory of relativity, we work µ l −→ µ j l with a model of spacetime, a four dimensional mani- (x ,x k) x ,a kx j. fold M endowed with a Lorentz metric (gµν ), such that L In this sense, P(M) is a principal fiber bundle over M the structure group of the orthonormal frame bun- O M L with its structure group and canonical fiber GL(n). dle ( ) can be reduced to the Lorentz subgroup 0, that is, there is a global smooth section of the bundle The Lorentz metric (gµν ) determines a fiber sub- O M L . Suppose s : M → O M L is such a sec- bundle O(M) consisting of all orthonormal frames ( )/ 0 ( )/ 0 tion which determines a spacetime orientation of M. (x,(e )) where g(e ,e ) = η . O(M) possesses a k σ ρ σρ → differential structure as the sub-manifold of dimension Suppose F : M M is a diffeomorphism preserving 10, defined by the following ten (independent) equa- the gravitational field (gµν ), so that the tangent map- tions: ping F⋆ sends the fiber O(M)x to the fiber O(M)F(x)for µ ν every x ∈ M. Let s(x) = [e(x)] for every x ∈ M, repre- gµν (x)x σ x ρ = ησρ . (1) sented by the equivalent class of a frame (x,(eµ (x))), j e x F e x O M It is easy to verify that the right action under (a k) and let ˜( ) = ⋆( ( )) which belongs to ( )F(x), σ ρ leaves O(M) invariant if and only if a η a = η , wheree ˜µ (x)= F⋆ eµ (x) for every x ∈ M. We say an µ σρ ν µν  that is a is a Lorentz matrix. Therefore O(M) is a prin- isometry F is proper and orthochronous (with respect cipal fiber bundle with its structure group and typical to the spacetime orientation s), if [e˜(x)] = [e(F(x))] for fiber the L . every x, that is, the spacetime orientation is invariant If (x,(eα )) and (x,(e˜α )) are two orthonormal frames under the differential map F⋆. ν of TxM, thene ˜µ = Λ µ eν where Λ is a Lorentz transfor- We are now in a position to formulate the prin- mation. It is said two frames (x,(eα )) and (x,(e˜α )) are ciple of Lorentz invariance: laws of Nature are in- ν L equivalent ife ˜µ = Λ µ eν with Λ ∈ 0. Thus, for every variant under any proper and orthochronous isometry x ∈ M, the fiber O(M)x is decomposed into a direct of (M,gµν ). That is, laws of Nature take the same sum of 4 disjoint connected components: O(M)x = mathematical formulation under any proper and or- [e] ∪ [Pe] ∪ [Te] ∪ [PTe] where e ∈ O(M)x (where P thochronous isometry of (M,gµν ). and T denote the space and time inversions respec- We conclude this note by raising the following ques- tively), which in turn leads to two principal bundles. tion, which should be worthy of study. Recall that Let O(M)/L0 denote the of equivalence classes, Einstein’s field equation and the geometric dynamics and the natural projection is denoted by σ : O(M) → were derived under the general version of Lorentz in- O(M)/L0 sending every element of O(M) to its equiv- variance, which is violated at the sub-atomic level. Of alent class. Then O(M) is a principal fiber bundle over course, there is no reason, though there is no any ex- O(M)/L0. On the other hand, the natural projection perimental evidence yet, to explain why the general π from O(M) to M defines a natural projection from Lorentz invariance is not violated at the level of geo-

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