JGRG 26 Conference, Osaka, Japan 24 October Frederic P. Schuller. Constructive gravity. Institute for Quantum Gravity Erlangen, Germany

Transcription

1 JGRG 26 Conference, Osaka, Japan 24 October 2016 Constructive gravity Frederic P. Schuller Institute for Quantum Gravity Erlangen, Germany

2 S matter ) S gravity JGRG 26 Conference, Osaka, Japan 24 October 2016 Constructive gravity Frederic P. Schuller Institute for Quantum Gravity Erlangen, Germany

3 THEORY

4 Usually postulate matter and gravity S = S [A,g] + S [g] universe matter gravity

5 Usually postulate matter and gravity S = S [A,g] + S [g] universe matter gravity matter fields

6 Usually postulate matter and gravity S = S [A,g] + S [g] universe matter gravity matter fields Lorentzian metric

7 Usually postulate matter and gravity S = S [A,g] + S [g] universe matter gravity matter fields Lorentzian metric

8 Usually postulate matter and gravity S = S [A,g] + S [g] universe matter gravity

9 This talk: Postulate only matter S = S [A,g] + S [g] universe matter gravity

10 This talk: Postulate only matter S = S [A,g] + S [g] universe matter gravity Does this suffice to derive the gravity action?

11 This talk: Postulate only matter S = S [A,g] + S [g] universe matter gravity Does this suffice to derive the gravity action? Before answering, let s raise the stakes even further

12 Admit matter on any background! S = S [A,g] + S [g] universe matter gravity

13 Admit matter on any background! S universe = S [A, G) + S [G] matter gravity

14 Admit matter on any background! S universe = S [A, G) + S [G] matter gravity any tensorial geometry e.g. G [ab][cd]

15 Admit matter on any background! S universe = S [A, G) + S [G] matter gravity any tensorial geometry e.g. G [ab][cd] So, can one derive the dynamics for G?

16 Yes. S = S [A, G) + S [G] universe matter gravity

17 Yes, for canonically quantizable matter. S = S [A, G) + S [G] universe matter gravity

18 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial

19 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime

20 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime 2 Legendre map

21 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime 2 Legendre map of matter initial data A,G

22 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime 2 Legendre map of matter initial data A,G of geometric initial data G

23 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime construction equations for the gravity Lagrangian 2 Legendre map 3 read off kinematical coefficients of matter initial data A,G of geometric initial data G

24 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime construction equations for the gravity Lagrangian 4 solve construction equations (hard) 2 Legendre map 3 read off kinematical coefficients of matter initial data A,G of geometric initial data G

25 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime construction equations for the gravity Lagrangian 4 solve construction equations (hard) 2 Legendre map 3 read off kinematical coefficients of matter initial data A,G of geometric initial data G

26 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial kinematics of underlying spacetime construction equations for the gravity Lagrangian 4 solve construction equations (hard) 2 Legendre map 3 read off kinematical coefficients of matter initial data A,G of geometric initial data G

27 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial 2011 kinematics of underlying spacetime Kuchar solve construction equations (hard) construction equations for the gravity Lagrangian 4 2 Legendre map read off kinematical coefficients of matter initial data A,G 1976 Hojman-Kuchar-Teitelboim of geometric initial data G

28 Yes, for canonically quantizable matter. S universe = S [A, G) + S [G] matter gravity 1 principal polynomial 2011 kinematics of underlying spacetime Kuchar solve construction equations (hard) construction equations for the gravity Lagrangian 4 2 Legendre map read off kinematical coefficients of matter initial data A,G 1976 Hojman-Kuchar-Teitelboim For all details see arxiv next week (Schuller et al.) of geometric initial data G

29 EXAMPLE

30 Example: Refined electrodynamics Given Z S matter [A; G] = 1 8 d 4 x! G G abcd F ab F cd what are the gravitational field equations for the geometry G? Since we can, we must not postulate

31 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial kinematics of underlying spacetime

32 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial kinematics of underlying spacetime

33 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial kinematics of underlying spacetime equations of motion

34 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial kinematics of underlying spacetime equations of motion ) principal polynomial P abcd G = 1 24! G 2 mnpq rstu G mnr(a G b sp c G d)qtu,

35 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime

36 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map of matter initial data A,G

37 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map of matter initial data A,G

38 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map of matter initial data A,G canonically quantizable, bi-hyperbolic P G

39 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map of matter initial data A,G canonically quantizable, bi-hyperbolic P G L G a (x, k) := P G am 2 m deg P (x) k m2 k mdeg P P G n 1 n deg P(x) kn1 k ndeg P

40 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map L G a of matter initial data A,G

41 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map L G a of matter initial data A,G use general construction result of geometric initial data G

42 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime Legendre map L G a of matter initial data A,G use general construction result of geometric initial data G

43 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

44 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

45 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a Use general rules for reading off coefficients read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

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47 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd principal polynomial P G (abcd) kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

48 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd S gravity [G] principal polynomial P G (abcd) solve construction equations kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

49 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd Set up construction equations S gravity [G] principal polynomial P G (abcd) solve construction equations kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

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52 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd Set up construction equations S gravity [G] principal polynomial P G (abcd) solve construction equations kinematics of underlying spacetime Legendre map L G a Solve construction equations for L gravity construction equations for the gravity Lagrangian read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

53 Only 6 constants to be determined by experiment

54 Example: Refined electrodynamics Z d 4 x! G G abcd F ab F cd + S gravity [G] principal polynomial P G (abcd) solve construction equations kinematics of underlying spacetime construction equations for the gravity Lagrangian Legendre map L G a read off kinematical coefficients of matter initial data A,G use general construction result of geometric initial data G

55 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing?

56 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing?

57 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 1. Solve the derived linearized gravitation field equation around point mass

58 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 1. Solve the derived linearized gravitation field equation around point mass subtly non-metric solution G 0 0 = G 0 = G = apple 1+ apple 1+ apple 1 (1 + )Mapple (3 + )Mapple 4 (1 + )Mapple e µr r e µr r e µr r M 32b M 16b 1 r 2 [ N ] 1 r ( )

59 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 1. Solve the derived linearized gravitation field equation around point mass subtly non-metric solution G 0 0 = G 0 = G = apple 1+ apple 1+ apple 1 (1 + )Mapple (3 + )Mapple 4 (1 + )Mapple e µr r e µr r e µr r M 32b M 16b 1 r 2 [ N ] 1 r ( ) parameters can be determined, in principle and for instance, by combined lensing and gravitational wave measurements

60 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 2. Calculate optical geometry (light rays) and magnification (first order WKB)

61 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 2. Calculate optical geometry (light rays) and magnification (first order WKB) Light rays exactly as for Einstein (!)

62 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 2. Calculate optical geometry (light rays) and magnification (first order WKB) Light rays exactly as for Einstein (!)

63 Application of this Example If there is bi-refringence of light in vacuo, what are the signatures in gravitational lensing? 2. Calculate optical geometry (light rays) and magnification (first order WKB) Light rays exactly as for Einstein (!) Subtle refinement to Einsteinian magnification, linear in the lens mass (!) = 1 + Mapple 16 apple e µd LO D LO e µdso D SO Result with M. Werner 2016

64 Conclusion Gravitational dynamics need not be postulated, since they can be d.

65 Conclusion The matter dynamics on a spacetime constructively determine the dynamics of the geometry.

66 Conclusion The construction of a modified gravity theory amounts to solving a countable set of linear homogeneous PDE.

67 Conclusion You can add your personally favored conditions on the gravity theory in order to further restrict the solution space.