Frolov: () koeppel@fias.uni-frankfurt.de Frankfurt Institute for Advanced Studies Institut für theoretische Physik Goethe-Universität Frankfurt Journal Club in High Energy Physics Mon. 15. Jun 2015, 16:00-17:00 1 / 18
Biswas 2013 Higher Derivative Gravity In this section Presenting the more readable results from Biswas, Gerwick, Koivisto, Mazumdar (arxiv:1110.5249, arxiv:1302.0532) Linearized g µν = η µν + h µν. The most general quadratic action: S q = d 4 x gr µ1 ν 1 λ 1 σ 1 D µ 1ν 1 λ 1 σ 1 µ 2 ν 2 λ 2 σ 2 R µ 2ν 2 λ 2 σ 2 Higher Derivative Gravity Confining the Lagrangian: 14 Terms Confining the Lagrangian: 14 Terms 3 Terms Confining the Lagrangian: 3 Terms with D {η µν, µ }. 2 / 18
Biswas 2013 Confining the Lagrangian: 14 Terms S q = d 4 x [ g RF 1 ( )R + RF 2 ( ) µ µ R µν + R µν F 3 ( )R µν + RµF ν 4 ( ) ν λ R µλ + R λσ F 5 ( ) µ σ ν λ R µν + RF 6 ( ) µ ν λ σ R µνλσ + R µλ F 7 ( ) ν σ R µνλσ + R ρ λ F 8( ) µ σ ν ρ R µνλσ + R µ1ν1 F 9 ( ) µ1 ν1 µ ν λ σ R µνλσ + R µνλσ F 10 ( )R µνλσ + R ρ µνλ F 11( ) ρ σ R µνλσ + R µρ1νσ 1 F 12 ( ) ρ1 σ1 ρ σ R µρνσ + R ν1ρ1σ1 µ F 13 ( ) ρ1 σ1 ν1 ν ρ σ R µνλσ + R µ1ν1ρ1σ1 F 14 ( ) ρ1 σ1 ν1 µ1 µ ν ρ σ R µνλσ].
Biswas 2013 Confining the Lagrangian: 14 Terms 3 Terms Using the antisymmetric properties of the Riemann tensor, R (µν)ρσ = R µν(ρσ) = 0 and the Jacobi identity α R µνβγ + γ R µνβα + β R µνγα Making 8 terms in GR. Linearizing: µ commute on Minkowski throwing out 3 terms due to symmetry Higher Derivative Gravity Confining the Lagrangian: 14 Terms Confining the Lagrangian: 14 Terms 3 Terms Confining the Lagrangian: 3 Terms 4 / 18
Biswas 2013 Confining the Lagrangian: 3 Terms Linearized Action: S q = d 4 x [RF 1 ( )R + R µν F 3 ( )R µν + R µνλσ F 10 ( )R µνλσ] can be casted in terms of h µν by expressions like R µνλσ = 1 2 ( [λ ν h µσ] [λ µ h νσ] ),..., R = ν µ h µν h yielding eq. (1) from the Frolov paper: S q = [ 1 d 4 x 2 h µν a( )h µν + hµb( ) σ σ ν h µν + hc( ) µ ν h µν + 1 2 f ( ) h d( )h + hλσ σ λ µ ν h µν]
The most general action S q = [ 1 d 4 x 2 h µν a( )h µν + hµb( ) σ σ ν h µν + hc( ) µ ν h µν + 1 2 GR Field equations can be used to show b = 0, c + d = 0, b + c + f = 0, with only two independent functions. f ( ) h d( )h + hλσ σ λ µ ν h µν]
General Realtivity (GR) Recover GR in infrared domain S q = [ 1 d 4 x 2 h µν a( )h µν + hµb( ) σ σ ν h µν + hc( ) µ ν h µν + 1 2 with a(0) = c(0) = b(0) = d(0) = 1 gives L = R f ( ) h d( )h + hλσ σ λ µ ν h µν]
Gauss-Bonnet (GB) Gauss-Bonnet term G = R 2 4R µν R µν + R µναβ R µναβ is a special case with a = c = 1, so that L = R + α( )G. (Interesting only in D 4 + 1 dimensions). The most general action General Realtivity (GR) Gauss-Bonnet (GB) L(R) Weyl Higher derivative (HD) Ghost free (GF) 8 / 18
L(R) Also called f (R) : Replace R in Einstein-Hilbert action with f (R), or: L(R) = L(0) + L (0)R + 1 2 L (0)R +... with a = 1, c = 1 L ( ). The most general action General Realtivity (GR) Gauss-Bonnet (GB) L(R) Weyl Higher derivative (HD) Ghost free (GF) 9 / 18
Weyl Using the Weyl tensor (traveless component of the Riemann tensor). Weyl tensor C µναβ = 0 when metric is (locally) conformally flat. Weyl-squared theory with 1/µ suppression: L = R 1 µ 2 C µναβc µναβ can be constructed with a = 1 µ 2, c = 1 1 3 µ 2 The most general action General Realtivity (GR) Gauss-Bonnet (GB) L(R) Weyl Higher derivative (HD) Ghost free (GF) 10 / 18
Higher derivative (HD) a = n (1 µ 2 i ) i=1 n c c = (1 ν 2 k ) k=1 i.e. a (in)finite amount of derivatives associated with energy scales. The most general action General Realtivity (GR) Gauss-Bonnet (GB) L(R) Weyl Higher derivative (HD) Ghost free (GF) 11 / 18
Ghost free (GF) a = c = exp( /µ 2 ) equal to NCG (Noncommutative Geometry) / NCBH (Noncommuting Black Holes), and similar to GUP. The most general action General Realtivity (GR) Gauss-Bonnet (GB) L(R) Weyl Higher derivative (HD) Ghost free (GF) 12 / 18
Static s of the linearized equations Start with point-source Energy-momentum tensor τ µν = ρδ 0 µδ 0 ν = mδ 3 ( r)δ 0 µδ 0 ν Compute metric in Newtonian limit ds 2 = (1 + 2Φ)dt 2 + (1 2Ψ)dx 2 by solving EFE (eg. geometric part:) κτ µ τ µ ν = 0 = (a+b) h µ ν,µ+(c+d) ν h+(b+c+f )h αβ Yields equations 2(a 3c)[ 2 Φ 4 2 Ψ] = κρ 2(c a) 2 Φ 4c 2 Ψ = κρ. We seek functions c( ) and a( ), such that there are no ghosts and no 1/r divergence at short distances Static s of the linearized equations The Comments,αβν 13 / 18
The With f = 0 ( non theory), 4a( 2 ) 2 Φ = 4a( 2 ) 2 Ψ = κρ = κmδ 3 ( r) For instance a( ) = e /M2 Fourier transformation of eq ( ) gives Φ(r) m dp /M 2 Mpr 2 p e p2 sin (p r) = like in String theory mπ 2M 2 p r erf Error function lim r erf(r) = 1 is GR limit Error function lim r 0 erf(r) = r makes Newtonian potential Φ(r) mm/m 2 p: Finite potential! Frolov: A finite is not neccessary regular one ; reproduces. ( ) rm 2 Static s of the linearized equations The Comments 14 / 18
Comments Evaluation The result Φ(r) is nothing new! The same linearized with ( ) ( ( ) ) rm rm 2 erf = γ 1/2; γ( 2 2 1 /2; r 2 /4θ) has already been done by Nicolini 2005 (arxiv:hep-th/0507266), where θ = 1/M is the NCBH fundamental length and therefore M is the minimal length. The same result was also recovered in 2005 by Gruppuso (arxiv:hep-th/0502144) with the erf representation. None of them were cited by Biswas, Gerwick, Koivisto, Mazumdar 2011 (arxiv:1110.5249). Static s of the linearized equations The Comments 15 / 18
Boosting the Object of investigation: Collapsing spherical thin null shell. To do so: Boost point source along y-axis with constant ultrarelativistic velocity β. After comoving transformation (t, y) (v, u), the field is found as Φ = 4GM F (ζ 2 ) δ(u) F (z) = 0 ds f (s) e z/(4s) s Remember: Only HD/GF has f 0. For example in GF, Boosting the Apparent horizon or not Conclusions F (z) z 1 4 z2 + O(z 3 ) (1) 16 / 18
Apparent horizon or not Author considers ominous set of photons added to the spacetime: ds 2 = ds 2 0 + dh2 Determine the position of an apparent horizon by 0 = ( g θθ ) 2 = ( (r 2 GM r zf (z))) 2 Find: For small enough mass M there is no apparent horizon! Mass gap for mini-black hole. Funnily: The divergence is a result of the unphysical assumption, that the thickness of the null shell is zero, and for the collapse of the shell with finite thickness this divergence becomes softer or is absent. Boosting the Apparent horizon or not Conclusions 17 / 18
Conclusions Evaluation In comparison with our models, eg. GUP, holographic/ self-regular principles or NCBHs, Frolov s result is probably not suprising: He spent a great effort to discover the G-lump which is smaller than the minimal mass and has no event horizon. Such an example is displayed for NCBH-like metric components g 00 in the lower plot (blue line ˆ= mass gap gravitational potential). Boosting the Apparent horizon or not Conclusions 1.0 0.8 0.6 M 0.5M 0 0.4 g 00 0.2 M 1.0M 0 0.0 0.2 0.4 M 1.5M 0 0.0 0.5 1.0 1.5 2.0 2.5 r r 0 18 / 18