Hello! I have recently bought the book(adsbygoogle = window.adsbygoogle || []).push({}); The Principle of Relativity by Einstein (Along with Minkowski, Lorentz and Weyl).This book is simply a collection of papers published by Einstein (along with the other three scientists mentioned) concerning the development of Special and General Relativity. To the best of my knowledge it is in chronological order. Now, don't worry, I am not attempting to study the Theories of Relativity from the original papers published on them. I have already self-studied SR and GR by watching the excellent video lectures on YouTube by Leonard Susskind, and reading Sean Carroll's free notes on GR (Which have been published into a graduate textbook that is unfortunately too expensive for me at the moment). Now to my question.

In the paper by Einstein titledThe Foundation of the General Theory of Relativity.Section 15 (page 145 in the book), he mentionsthe Hamiltonian Function for the Gravitational Field.Which he says, "To show that the field equations correspond to the laws of momentum and energy, it is most convenient to write them in the following Hamiltonian form..." He then goes on to define the Hamiltonian for a gravitational field in the absence of matter as ##H=g^{\mu\nu}{\Gamma}^{\alpha}_{\mu\beta}{\Gamma}^{\beta}_{\nu\alpha}## along with the convention that ##(-g)^{1/2}=1## where ##g## is the determinant of the metric. To the best of my knowledge, the modern way to derive the Field Equations is to vary the action of the gravitational field, having the Lagrangian density for the gravitational field be ##R## the Ricci Scalar. I have looked everywhere to see if this form of the Hamiltonian for a non-quantum gravitational field (in the absence of matter) is used, and how valid is it. When looking up the Hamiltonian for Gravity I often get the ADM Formulation for Quantum Gravity, which isn't what I'm looking for as I would like a non-quantum formulation. My question(s), I guess, are;

1) Is this a correct Hamiltonian for the Gravitational Field?

2)If yes, where did Einstein get this form of the Hamiltonian from? Guess and Check? I doubt that, though.

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# Einstein's Hamiltonian?

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