The

Eisenhart-Lift

for Field Theories

K. Finn et al. 2018

arxiv:1806.02431

Are

Conservative Forces Fictitious?

K. Finn et al. 2018

arxiv:1806.02431

Title Text

Subtitle

Title Text

Subtitle

Newtonian dynamics in Euclidean space

GR

in space-time

Title Text

Subtitle

Newtonian dynamics in Euclidean space

GR

in space-time

Maxwell's equ.

in space-time

Kaluza-Klein theory

in 5D

Title Text

Subtitle

Newtonian dynamics in Euclidean space

GR

in space-time

Maxwell's equ.

in space-time

Kaluza-Klein theory

in 5D

Eisenhart-lift

L = \frac{1}{2}m\sum_{i=1}^{n} \dot{x}_i^2 + \frac{1}{2}\frac{M^2}{V(\mathbf{x})} \dot{y}^2
L = \frac{1}{2}m\sum_{i=1}^{n} \dot{x}_i^2 - V(\mathbf{x})

Eisenhart-Lift

\mathcal{L} = \sqrt{-g} \left( \frac{1}{2}g^{\mu\nu}k_{ij}(\mathbf{\phi})\partial_\mu\phi^i\partial_\nu\phi^j + \frac{1}{2}\frac{M^4}{V(\mathbf{\phi})} \nabla_\mu B^\mu \nabla_\nu B^\nu \right)
\mathcal{L} = \sqrt{-g} \left( \frac{1}{2}g^{\mu\nu}k_{ij}(\mathbf{\phi})\partial_\mu\phi^i\partial_\nu\phi^j - V(\mathbf{\phi})\right)

field theory with a potential

free theory with extra d.o.f.

Conclusion

  • Any scalar field theory can be written in a kinetic-only form

Application

  • Multi-field inflation
  • Initial conditions in inflation (arXiv:1812.07095)
  • Using geometric reasoning for field theory problems