- #36
WannabeNewton
Science Advisor
- 5,844
- 551
Take the following example: Consider an observer [itex]S[/itex] and another observer [itex]S'[/itex] who set up respective coordinates [itex]x^{\mu },x'^{\mu}[/itex] on some open subset [itex]U\subseteq M[/itex] where [itex](M,g_{ab})[/itex] is a space - time. For each [itex]p\in U[/itex], the observers use their respective coordinate bases [itex]\left \{ \frac{\partial }{\partial x^{i}}|p \right \}_{i},\left \{ \frac{\partial }{\partial x'^{i}}|p \right \}_{i} [/itex] for [itex]T_{p}(M)[/itex]. Say we have a rank 2 tensor [itex]T\in T_p(M)\otimes T_p(M)[/itex] representing a physically measurable quantity. In this passive point of view, what general covariance tells us is that if [itex]S[/itex] makes a measurement, using the measuring apparatus used to define his local coordinates, of the physical quantity represented by [itex]T[/itex] and [itex]S'[/itex] does the same then the values [itex]T^{\mu \nu }[/itex] obtained in measurement by [itex]S[/itex] will relate to the values [itex]T'^{\mu \nu }[/itex] measured by [itex]S'[/itex] by [itex]T'^{\mu \nu } = \frac{\partial x'^{\mu }}{\partial x^{\alpha}}\frac{\partial x'^{\nu}}{\partial x^{\beta}}T^{\alpha\beta}[/itex]. If in particular, we instead talk about the value of a scalar field at a point then such observers, [itex]S,S'[/itex] will measure the same exact value for the value of the scalar field at that point.
The active point of view, instead of looking at coordinates and coordinate transformations on a given space - time, looks at the image of the space - time under the diffeomorphism and the pull back and pushforward, under this diffeomorphism, of tensor fields representing physically measurable quantities and again asserts general covariance.
The active point of view, instead of looking at coordinates and coordinate transformations on a given space - time, looks at the image of the space - time under the diffeomorphism and the pull back and pushforward, under this diffeomorphism, of tensor fields representing physically measurable quantities and again asserts general covariance.