Higher dimensional massive (bi-)gravity: Constructions and solutions - PowerPoint PPT Presentation

Higher dimensional massive (bi-)gravity: Constructions and solutions Tuan Q. Do Vietnam National University, Hanoi Based on PRD93(2016)104003 [arXiv:1602.05672]; PRD94(2016)044022 [arXiv:1604.07568]. Hot Topics in General Relativity and Gravitation 3 (HTGRG-3) XIIIth Rencontres du Vietnam ICISE, Quy Nhon, July 30th - August 5th, 2017 1 / 26

Contents Motivations 1 Cayley-Hamilton theorem and ghost-free graviton terms 2 Simple solutions for a five-dimensional massive gravity 3 Simple solutions for a five-dimensional massive bi-gravity 4 Conclusions 5 2 / 26

I. Motivations The massive gravity [gravitons have tiny but non-zero mass] has had a long and rich history since the seminal paper by Fierz & Pauli [PRSA173(1939)211] . van Dam & Veltman [NPB22(1970)397] and Zakharov [PZETF12(1970)447] showed that in the massless limit, it cannot recover GR. Vainshtein pointed out that the nonlinear extensions of FP theory can solve the vDVZ discontinuity problem [PLB39(1972)393] . Boulware & Deser claimed that there exists a ghost associated with the sixth mode in graviton coming from nonlinear levels [PRD6(1972)3368] . Building a ghost-free nonlinear massive gravity, in which a massive graviton carries only five ”physical” degrees of freedom, has been a great challenge for physicists. de Rham, Gabadadze & Tolley (dRGT) have successfully constructed a ghost-free nonlinear massive gravity [1011.1232, 1007.0443] . The dRGT theory has been proved to be ghost-free for general fiducial metric by some different approaches, e.g., Hassan & Rosen [1106.3344, 1109.3230] . The dRGT theory might be a solution to the cosmological constant problem. 3 / 26

I. Motivations An interesting extension of dRGT theory is the massive bi-metric gravity (bi-gravity) proposed by Hassan & Rosen, in which the reference metric is introduced to be full dynamical as the physical metric [1109.3515] . For interesting review papers, see de Rham [1401.4173] ; K. Hinterbichler [1105.3735] ; Schmidt-May & von Strauss [1512.00021] . It is noted that most of previous papers have focused only on four-dimensional frameworks, which involve only the first three massive graviton terms, L 2 , L 3 , and L 4 . There have been a few papers discussing higher dimensional scenarios of massive (bi)gravity theories, e.g., Hinterbichler & Rosen [1203.5783] ; Hassan, Schmidt-May & von Strauss [1212.4525] ; Huang, Zhang & Zhou [1306.4740] . However, these papers have not studied the well-known metrics in higher dimensions, e.g., the Friedmann-Lemaitre-Robertson-Walker (FLRW), Bianchi type I, and Schwarzschild-Tangherlini metrics. We would like to investigate whether the five-dimensional (bi)gravity theories admit the above metrics as their solutions. 4 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms Recall the four-dimensional action of the dRGT massive gravity [1011.1232, 1007.0443] : S 4d = M 2 d 4 x √− g � � � p R + m 2 g ( L 2 + α 3 L 3 + α 4 L 4 ) , 2 where M p the Planck mass, m g the graviton mass, α 3 , 4 free parameters, and the massive graviton terms L i defined as L 2 = [ K ] 2 − [ K 2 ]; L 3 = 1 3[ K ] 3 − [ K ][ K 2 ] + 2 3[ K 3 ] , L 4 = 1 12[ K ] 4 − 1 2[ K ] 2 [ K 2 ] + 1 4[ K 2 ] 2 + 2 3[ K ][ K 3 ] − 1 2[ K 4 ] . Square brackets: ν ) 2 ; [ K 2 ] ≡ tr K µ ν ; [ K ] 2 ≡ (tr K µ [ K ] ≡ tr K µ α K α ν ; and so on . The square matrix K µν is defined as � K µ ν ≡ δ µ f ab ∂ µ φ a ∂ α φ b g αν , ν − φ a ∼ St¨ uckelberg fields; g µν ∼ (dynamical) physical metric , f ab ∼ non-dynamical reference (fiducial) metric of massive gravity . 5 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms Recall the four-dimensional action of the massive bi-gravity [1109.3515] : √ � d 4 x √ gR ( g ) + M 2 � S 4d = M 2 d 4 x f R ( f ) g f � d 4 x √ g � � + 2 m 2 M 2 U 2 + α 3 U 3 + α 4 U 4 , eff where � 1 � − 1 U i = 1 + 1 2 L i ; M 2 eff ≡ . M 2 M 2 g f The square matrix K µν is defined as K µ ν ≡ δ µ � f µα g αν , ν − g µν ∼ (dynamical) physical metric , f µν ∼ full dynamical reference (fiducial) metric . 6 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms We will construct higher dimensional terms L n > 4 by applying the well-known Cayley-Hamilton theorem for the square matrix K µν . In algebra, there exists the well-known Cayley-Hamilton theorem: any square matrix must obey its characteristic equation . In particular, given a n × n matrix K with its characteristic equation, P ( λ ) ≡ det( λ I n − K ) = 0, then P ( K ) ≡ K n − D n − 1 K n − 1 + D n − 2 K n − 2 − ... +( − 1) n − 1 D 1 K + ( − 1) n det( K ) I n = 0 , where D n − 1 = tr K ≡ [ K ] and D n − j (2 ≤ j ≤ n − 1) are coefficients of the characteristic polynomial. For n = 2, the following characteristic equation: K 2 − [ K ] K + det K 2 × 2 I 2 = 0 , which implies after taking the trace det K 2 × 2 = 1 ∼ L 2 � [ K ] 2 − [ K 2 ] � 2 . 2 7 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms For n = 3, the corresponding characteristic equation: K 3 − [ K ] K 2 + 1 [ K ] 2 − [ K 2 ] � � K − det K 3 × 3 I 3 = 0 , 2 which leads to det K 3 × 3 = 1 ∼ L 3 � [ K ] 3 − 3[ K 2 ][ K ] + 2[ K 3 ] � 2 . 6 For n = 4, the corresponding characteristic equation: K 4 − [ K ] K 3 + 1 [ K ] 2 − [ K 2 ] K 2 � � 2 − 1 [ K ] 3 − 3[ K 2 ][ K ] + 2[ K 3 ] � � K + det K 4 × 4 I 4 = 0 , 6 which gives det K 4 × 4 = 1 ∼ L 4 [ K ] 4 − 6[ K ] 2 [ K 2 ] + 3[ K 2 ] 2 + 8[ K ][ K 3 ] − 6[ K 4 ] � � 2 . 24 The higher dimensional graviton terms L n > 4 must vanish in all four-dimensional spacetimes. 8 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms The higher dimensional terms L n > 4 = det K n × n / 2 can be constructed from the Cayley-Hamilton theorem to be L 5 1 � [ K ] 5 − 10[ K ] 3 [ K 2 ] + 20[ K ] 2 [ K 3 ] 2 = 120 − 20[ K 2 ][ K 3 ] + 15[ K ][ K 2 ] 2 − 30[ K ][ K 4 ] + 24[ K 5 ] � , L 6 1 � [ K ] 6 − 15[ K ] 4 [ K 2 ] + 40[ K ] 3 [ K 3 ] − 90[ K ] 2 [ K 4 ] 2 = 720 + 45[ K ] 2 [ K 2 ] 2 − 15[ K 2 ] 3 + 40[ K 3 ] 2 − 120[ K 3 ][ K 2 ][ K ] � + 90[ K 4 ][ K 2 ] + 144[ K 5 ][ K ] − 120[ K 6 ] , L 7 1 � [ K ] 7 − 21[ K ] 5 [ K 2 ] + 70[ K ] 4 [ K 3 ] − 210[ K ] 3 [ K 4 ] 2 = 5040 + 105[ K ] 3 [ K 2 ] 2 − 420[ K ] 2 [ K 2 ][ K 3 ] + 504[ K ] 2 [ K 5 ] − 105[ K 2 ] 3 [ K ] + 210[ K 2 ] 2 [ K 3 ] − 504[ K 2 ][ K 5 ] + 280[ K 3 ] 2 [ K ] − 420[ K 3 ][ K 4 ] � + 630[ K 4 ][ K 2 ][ K ] − 840[ K 6 ][ K ] + 720[ K 7 ] . 9 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms A five-dimensional scenario of massive gravity [1602.05672] : S = M 2 � d 5 x √− g � � p R + m 2 g ( L 2 + α 3 L 3 + α 4 L 4 + α 5 L 5 ) , 2 The corresponding five-dimensional Einstein field equations: � R µν − 1 � + m 2 2 Rg µν g ( X µν + σ Y µν + α 5 W µν ) = 0 , X µν = − 1 2 ( α L 2 + β L 3 ) g µν + ˜ X µν , ˜ � K 2 � X µν = K µν − [ K ] g µν − α µν − [ K ] K µν � µν + L 2 � K 3 µν − [ K ] K 2 + β 2 K µν , Y µν = − L 4 Y µν = L 3 2 K µν − L 2 2 g µν + ˜ Y µν ; ˜ 2 K 2 µν + [ K ] K 3 µν − K 4 µν , W µν = − L 5 2 g µν + ˜ W µν , W µν = L 4 2 K µν − L 3 µν + L 2 ˜ 2 K 2 2 K 3 µν − [ K ] K 4 µν + K 5 µν , 10 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms Here α = α 3 + 1, β = α 3 + α 4 , and σ = α 4 + α 5 . Note that Y µν = 0 in four dimensional spacetimes [Do & Kao, PRD88(2013)063006] but � = 0 in higher-than-four dimensional ones . Similarly, W µν = 0 in five dimensional spacetimes but � = 0 in higher-than-five dimensional ones. The constraint equations associated with the existence of fiducial metric: W µν − 1 t µν ≡ ˜ X µν + σ ˜ Y µν + α 5 ˜ 2 ( α 3 L 2 + α 4 L 3 + α 5 L 4 ) g µν = 0 . Due to these constraint equations the Einstein field equations for g µν become 2 Rg µν ) − m 2 ( R µν − 1 g 2 L M g µν = 0; L M ≡ L 2 + α 3 L 3 + α 4 L 4 + α 5 L 5 , ⇒ ( R µν − 1 2 Rg µν ) + Λ M g µν = 0 (Bianchi constraint , ∂ ν L M = 0) , with Λ M ≡ − m 2 g L M / 2 as an effective cosmological constant. 11 / 26

II. Cayley-Hamilton theorem and ghost-free graviton terms Ghost free issue Follow the analysis of dRGT papers [1011.1232, 1007.0443] by considering the tensor X ( n ) µν and its the recursive relation: n n ! � µν L ( n − m ) X ( n ) ( − 1) m 2( n − m )! K m µν ( g µν , K ) = ( K ) der m =0 + K αβ X ( n − 1) X ( n ) µν = − n K α µ X ( n − 1) g µν . αν αβ For the 4D case X (4) µν ( g µν , K ) ∼ Y µν = 0 → X ( n > 4) ( g µν , K ) = 0 → no µν ghostlike pathology arises at the quartic or higher order levels with arbitrary physical and fiducial metrics. Similarly, for the 5D case X (5) µν ( g µν , K ) ∼ W µν = 0 → X ( n > 5) ( g µν , K ) = 0 → µν any ghostlike pathology arising at the quintic or higher order levels must disappear, no matter the form of physical and fiducial metrics. The similar conclusion is also valid for higher-than-five massive gravity theories. 12 / 26