The Geometric Scalar Gravity Theory

Transcription

1 The Geometric Scalar Gravity Theory M. Novello 1 E. Bittencourt 2 J.D. Toniato 1 U. Moschella 3 J.M. Salim 1 E. Goulart 4 1 ICRA/CBPF, Brazil 2 University of Roma, Italy 3 University of Insubria, Italy 4 University of Cambridge, United Kingdom II JPB Cosmology School Pedra Azul, Domingos Martins - ES, 2014

2 Outline 1 Introduction Relevant scalars theories GSG s basics properties 2 Motivation 3 Developing the theory 4 Final Comments

3 Relevant scalars theories Nordström theory In 1912 Nordström achieved the first well acceptable generalization of Newton s gravitation.

4 Relevant scalars theories Nordström theory In 1912 Nordström achieved the first well acceptable generalization of Newton s gravitation. Field equation Φ Φ = κ T (1)

5 Relevant scalars theories Nordström theory In 1912 Nordström achieved the first well acceptable generalization of Newton s gravitation. Field equation Φ Φ = κ T (1) Test particles moving according with d dτ (m vµ ) = m η µν νφ (2)

6 Relevant scalars theories Nordström theory In 1912 Nordström achieved the first well acceptable generalization of Newton s gravitation. Field equation Φ Φ = κ T (1) Test particles moving according with d dτ (m vµ ) = m η µν νφ (2) Mass was no longer a constant m = m 0 Φ

7 Relevant scalars theories Einstein-Fokker s theory A few years later, in 1914, the previous theory is reformulated in a geometric way.

8 Relevant scalars theories Einstein-Fokker s theory A few years later, in 1914, the previous theory is reformulated in a geometric way. Field equation becomes R = 24πG T (3)

9 Relevant scalars theories Einstein-Fokker s theory A few years later, in 1914, the previous theory is reformulated in a geometric way. Field equation becomes R = 24πG T (3) Test particle follows geodesics in the curved space with metric g µν = Φ 2 η µν (4)

10 Relevant scalars theories Einstein-Fokker s theory A few years later, in 1914, the previous theory is reformulated in a geometric way. Field equation becomes R = 24πG T (3) Test particle follows geodesics in the curved space with metric g µν = Φ 2 η µν (4) Although this is essentially the same theory that of Nordström, it represents the first time that gravitational interactions has a purely geometric description.

11 Relevant scalars theories Basic problems in scalar gravity Scalar gravity deals with essentially three assumptions: The source of the gravitational field is the trace of the energy-momentum tensor Conformally flat geometry Violation of the relativity of inertia

12 Relevant scalars theories Basic problems in scalar gravity Scalar gravity deals with essentially three assumptions: No bending of light! Conformally flat geometry Violation of the relativity of inertia

13 Relevant scalars theories Basic problems in scalar gravity Scalar gravity deals with essentially three assumptions: No bending of light! Minkowski s background is observable! Violation of the relativity of inertia

14 Relevant scalars theories Basic problems in scalar gravity Scalar gravity deals with essentially three assumptions: No bending of light! Minkowski s background is observable! Preferred coordinate systems!

15 Relevant scalars theories Basic problems in scalar gravity Scalar gravity deals with essentially three assumptions: No bending of light! Minkowski s background is observable! Preferred coordinate systems! = Geometric Scalar Gravity (GSG) does not take in to account these assumptions, overcoming the attendant problems.

16 GSG s basics properties Fundamental properties of GSG 1 We assume that gravity is described by a Riemannian geometry.

17 GSG s basics properties Fundamental properties of GSG 1 We assume that gravity is described by a Riemannian geometry. 2 The gravitational metric is determined by derivatives of the gravitational field Φ, namely q µν = a η µν + b µφ νφ. (5)

18 GSG s basics properties Fundamental properties of GSG 1 We assume that gravity is described by a Riemannian geometry. 2 The gravitational metric is determined by derivatives of the gravitational field Φ, namely q µν = a η µν + b µφ νφ. (5) 3 Thus, all kind of matter and energy interact with Φ only through q µν.

19 GSG s basics properties Fundamental properties of GSG 1 We assume that gravity is described by a Riemannian geometry. 2 The gravitational metric is determined by derivatives of the gravitational field Φ, namely q µν = a η µν + b µφ νφ. (5) 3 Thus, all kind of matter and energy interact with Φ only through q µν }{{} Fundamental Hypothesis.

20 GSG s basics properties Fundamental properties of GSG 1 We assume that gravity is described by a Riemannian geometry. 2 The gravitational metric is determined by derivatives of the gravitational field Φ, namely q µν = a η µν + b µφ νφ. (5) 3 Thus, all kind of matter and energy interact with Φ only through q µν }{{} Fundamental Hypothesis. Example: Electromagnetic field The electromagnetic part of the Lagrangian is given by F = F αµf βν q αβ q µν. The corresponding field equation obtained by the variational principle δ 1 q F d 4 x = 0, (6) 4 is given by F µν ;ν = 0, (7) where the semicolon represents the covariant derivative w.r.t. the gravitational metric.

21 Motivation Hidden geometries in non linear scalar field theory Our main inspiration comes from the following paper,

22 Motivation Hidden geometries in non linear scalar field theory Our main inspiration comes from the following paper, Main statement The dynamics of a scalar field endowed with a Lagrangian L(w, ϕ) in a given background can be mapped in an different spacetime in which its metric is constructed by the field ϕ itself (and it derivatives).

23 Motivation An specific geometrization of a scalar field We consider the following Lagrangian in flat Minkowski spacetime, where w = η µν µφ νφ. The field equations is L = V (Φ) w, (8) 1 ( µ η η µν ) νφ + 1 V η 2 V w = 0. (9)

24 Motivation An specific geometrization of a scalar field We consider the following Lagrangian in flat Minkowski spacetime, where w = η µν µφ νφ. But, L = V (Φ) w, (8) 1 ( µ η η µν ) νφ + 1 V w Φ, (9) η 2 V where the represents the d Alambertian operator calculated in terms of the metric, q µν = α(φ) η µν + β(φ) w µ Φ ν Φ, (10) with α + β = α 3 V. (11)

25 Motivation An specific geometrization of a scalar field We consider the following Lagrangian in flat Minkowski spacetime, where w = η µν µφ νφ. But, L = V (Φ) w, (8) 1 ( µ η η µν ) νφ + 1 V w Φ, (9) η 2 V where the represents the d Alambertian operator calculated in terms of the metric, q µν = α(φ) η µν + β(φ) w µ Φ ν Φ, (10) with α + β = α 3 V. (11) Thus, the dynamic of Φ in the curved spacetime, given by q µν, is Φ = 0. (12)

26 Specifying the coefficients α and the potential V First, we consider the weak field regime in the case of a test particle, in order to have Newtonian limit for the theory, we have that d 2 x i dt 2 = Γi 00 = i Φ N. (13)

27 Specifying the coefficients α and the potential V First, we consider the weak field regime in the case of a test particle, in order to have Newtonian limit for the theory, we have that d 2 x i dt 2 = Γi 00 = i Φ N. (13) Using the metric definition to calculated this connection component, the following holds 1 α 1 + 2Φ N. (14)

28 Specifying the coefficients α and the potential V First, we consider the weak field regime in the case of a test particle, in order to have Newtonian limit for the theory, we have that d 2 x i dt 2 = Γi 00 = i Φ N. (13) Using the metric definition to calculated this connection component, the following holds 1 α 1 + 2Φ N. (14) We decide to explore the consequences of extrapolating this expression and define α = e 2Φ. (15)

29 Specifying the coefficients α and the potential V First, we consider the weak field regime in the case of a test particle, in order to have Newtonian limit for the theory, we have that d 2 x i dt 2 = Γi 00 = i Φ N. (13) Using the metric definition to calculated this connection component, the following holds 1 α 1 + 2Φ N. (14) We decide to explore the consequences of extrapolating this expression and define α = e 2Φ. (15) Now, it only remains to determine the potential.

30 Being guided by the observations To select among all possible V (Φ), we look from the indications from the various circumstances in which reliable experiments have been performed.

31 Being guided by the observations To select among all possible V (Φ), we look from the indications from the various circumstances in which reliable experiments have been performed. We then look for the static and spherically symmetric case, in order to determine V in such way that this solution be satisfactory in a observational way.

32 Being guided by the observations To select among all possible V (Φ), we look from the indications from the various circumstances in which reliable experiments have been performed. We then look for the static and spherically symmetric case, in order to determine V in such way that this solution be satisfactory in a observational way. So, assuming Φ = Φ(r), where r is the radial coordinate, the gravitational metric takes the form ds 2 = 1 α dt2 1 ( α 2 1 r dφ ) dr 2 r 2 dω 2. (16) V dr

33 Being guided by the observations To select among all possible V (Φ), we look from the indications from the various circumstances in which reliable experiments have been performed. We then look for the static and spherically symmetric case, in order to determine V in such way that this solution be satisfactory in a observational way. So, assuming Φ = Φ(r), where r is the radial coordinate, the gravitational metric takes the form ds 2 = 1 α dt2 1 ( α 2 1 r dφ ) dr 2 r 2 dω 2. (16) V dr In order to have a Schwarzschild-like geometry, we impose q 11 = 1/q 00. Applying in the dynamical equation Φ = 0, we get Φ(r) = 1 2 ln ( 1 rh r ), V = with r H = 2MG/c 2, and the final metric will be ( ds 2 = 1 rh r ) ( dt 2 1 rh r (3 α)2 4α 3. (17) ) 1 dr 2 r 2 dω 2. (18)

34 Final theory Now the theory is complete. The dynamic of the gravitational field is given by the Lagrangian L = V (Φ) w, (19) where the potential is with V = (3 α)2 4α 3, (20) α = e 2Φ. (21) The gravitational metric generated by Φ is q µν = α η µν + β µ Φ ν Φ w where the following expression determines the value of β,, (22) α + β = α 3 V. (23)

35 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic.

36 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. The action of the scalar field is easily described in the curved spacetime, η q S Φ = V w d 4 x = Ω V d 4 x, with Ω q αβ αφ β Φ. (24)

37 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. The action of the scalar field is easily described in the curved spacetime, η q S Φ = V w d 4 x = Ω V d 4 x, with Ω q αβ αφ β Φ. (24) From the variational principle, we get q δ S Φ = 2 V Φ δφ d 4 x. (25)

38 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x.

39 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Action corresponding to the matter, q S m = Lmd 4 x, (24) which returns that, δs m = 1 2 q T µν δq µν d 4 x, where T µν 2 q δ( q L m) δq µν. (25)

40 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Action corresponding to the matter, q S m = Lmd 4 x, (24) which returns that, δs m = 1 2 q T µν δq µν d 4 x, where T µν 2 q δ( q L m) δq µν. (25) But the metric q µν is not the fundamental quantity. We have to vary it as a function of Φ.

41 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: [( ) ] q 3 + α δs m = E T C λ ;λ δφ d 4 x, (24) 3 α T T µν q µν, E T µν µφ νφ, Ω (25) C λ β ( T λµ E q λµ) µφ α Ω (26)

42 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: [( q 3 + α δs m = 3 α ) ] E T C λ ;λ } {{ } 2 χ δφ d 4 x, (24) T T µν q µν, E T µν µφ νφ, Ω (25) C λ β ( T λµ E q λµ) µφ α Ω (26)

43 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: = δs m = 2 q χ δφ d 4 x,

44 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: = δs m = 2 q χ δφ d 4 x, Thus, the equation of motion takes the form V Φ = κ χ. (24)

45 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: = δs m = 2 q χ δφ d 4 x, Thus, the equation of motion takes the form V Φ = κ χ. (24) The quantity χ involves a non-trivial coupling between the gradient of the scalar field and the energy-momentum tensor of the matter that allows the electromagnetic field to interact with the gravitational field.

46 Action principle Once we have set the theory s potential and the coefficients of the gravitational metric, we proceed to description of its full dynamic. Scalar variational principle: = δs Φ = 2 q V Φ δφ d 4 x. Matter variational principle: = δs m = 2 q χ δφ d 4 x, Thus, the equation of motion takes the form V Φ = κ χ. (24) The quantity χ involves a non-trivial coupling between the gradient of the scalar field and the energy-momentum tensor of the matter that allows the electromagnetic field to interact with the gravitational field. The Newtonian limit gives the identification κ 8πG c 4. (25)

47 Final comments By assuming Einstein s idea, that gravity is a metrical phenomenon, and using the geometrization technique for a scalar field, we showed that is possible to construct a gravitational theory that overcomes the traditional drawbacks existing for the old scalar theories of gravity. In a first moment, we choose to determine all theory s parameters using the Newtonian limit and the solar system as a reference. Our results was published in JCAP 06(2013) 014. Some important issues are already being investigated by our co-workers, such as Cosmology (Toniato and Novello) Gravitational waves (Moschella and Bittencourt) Interior solutions (Rua and Novello)

48 THANKS!