𝔸𝕟 𝕖𝕢𝕦𝕚𝕧𝕒𝕝𝕖𝕟𝕥 𝕗𝕠𝕣𝕞𝕦𝕝𝕒𝕥𝕚𝕠𝕟 𝕠𝕗 𝕘𝕖𝕟𝕖𝕣𝕒𝕝 𝕣𝕖𝕝𝕒𝕥𝕚𝕧𝕚𝕥𝕪 𝕨𝕚𝕥𝕙𝕠𝕦𝕥 𝕘𝕖𝕠𝕞𝕖𝕥𝕣𝕪

One of the commonly cited concise definitions of general relativity is: matter tells spacetime how to curve, and spacetime tells matter how to move. Starting from this definition, we can outline a plan for a non-geometric formulation of general relativity by removing the intermediary and letting matter directly tell objects how to move. This can be achieved if we arrive at a complete form of the laws of conservation of energy and momentum for a body. This has in fact already been accomplished, as expressed by the law shown in its mathematical form in the figure, where energy and momentum in a gravitational field are redefined by multiplying the traditional definition by the metric and adding another term representing the gravitational field. When we substitute the values of these quantities from Schwarzschild spacetime, we obtain a simpler expression. The new relations can be illustrated by a simple geometric model, where each quantity is represented by a line segment in a simple diagram.