Energymomentum current of gravity in FinslerKawaguchi Lagrangian formulation

Energy-momentum current of gravity in Finsler-Kawaguchi Lagrangian formulation Ryoko Yahagi (Ochanomizu Univ. ) 矢萩　量子 Collaborators Takayoshi Ootsuka (Ochanomizu Univ. ) Muneyuki Ishida (Meisei Univ. ) Erico Tanaka (Kagoshima Univ. ) 1406. 2147 [gr-qc] To appear in CQG

Lagrangian Formalism based on Variational Principle Geometrical Lagrangians?

Relativistic free particle Nambu-Goto action World sheet World line Length Area Reparametrization invariance Conservation law of energy/energy-momentum current appears in Euler-Lagrange equations

Lagrangian Reparametrization invariance Energy-momentum current Choose the best space “ ” for your Lagrangian w Ne. Application to general relativity

Finsler geometry manifold metric cf. Riemannian geometry length of the curve

Finsler geometry extension of Riemannian geometry manifold Finsler metric Riemannian case: length of the curve

Kawaguchi geometry manifold Kawaguchi metric length of the curve A. Kawaguchi, Period Math Hung, 7, 291 (1976)

Kawaguchi geometry (areal space) manifold Kawaguchi metric area of the 2 -dim submanifold A. Kawaguchi, Bull Calcutta Math. Soc. , 56, 91,

　 　 Riemannian 　 　 1 st order Finsler 2 nd order Kawaguchi 　 　 Kawaguchi 　　　 (areal space) 　　　・ 　 　　　　　・ 　　　　　　　・ ・・・ 　 　 ・ ・ ・ 　 　 ・ ・ Reparametrization ・ 　 　 invariant 　 　 　

Langrangian Manifold Dynamical systems Finsler manifold + Metric Length

Field theories Kawaguchi manifold Area

　 　 1 st order Riemannian Finsler 　 　 　 ・ ・ ・ 2 nd order Kawaguchi ・・・ Dynamical systems 　 　 Kawaguchi 　　　・ 　　　　　　　・ 　 　 　 Field 　 theories 　 　 ・・・ 　 General relativity 　 　

Scalar Field (1+1 dim) = 0 EOM Conservation law of the energy-momentum

Pull-back

General relativity : cosmological const. Problem energy-momentum tensor of matter Not a conservation law! (general coordinate transformation) Not gauge invariant! energy-momentum pseudo-tensor of gravity pseudo

General relativity 2 nd order Kawaguchi manifold

EOM 0 = Energy-momentum current Pull-back of EOM Einstein eq.

Gauge transformation (general coordinate transformation) : functions of Generator: On-shell Einstein eq. ! t n a i ar v n i e g u a G

Open Question Pull-back of Einstein eq. On-shell !?

for Schwarzschild metric

for Schwarzschild metric negative! On-shell :

Conclusion Lagrangian Reparametrization invariance Energy-momentum current Finsler-Kawaguchi geometry Energy-momentum current of gravity