# Generalization of special relativity to curved space

Special relativity ignores curvature of space-time due to presence of mass and energy which is discussed in GR of course. in a curved space a physicist also can live and have true electrodynamics but ignoring curvature of space-time due to mass and energy as did Maxwell and Einstein in 1905 about our maybe flat space. what is this electrodynamics (which is core to special relativity) and it's corresponding special relativity? I mean main axioms, generalization of main definitions and main equations.

• It's slightly unclear what you want to ask. Do you want to ask what is the generalization of special relativistic electrodynamics in general relativity (i.e. in curved spacetime)? – Dvij Mankad Jun 27 '17 at 0:20
• You can do special relativity on any Lorentzian manifold, not just Minkowski spacetime. The equations are just the expected covariant versions of the classical ones, with pseudo-metric $g$ explicitly featured, see Cabral-Lobo's paper. If expressed invariantly there is no change at all. – Conifold Jun 27 '17 at 0:44

Your level is not altogether clear, but in any case you can read the following as saying that there is a very natural way to think of these things.

Electrodynamics in any geometry is most clearly written in the notation of exterior calculus and differential forms:

$$\begin{array}{lcl}\mathrm{d} \,F &=& 0\\\mathscr{Y}_0\,\mathrm{d}\star F &=& \star J\end{array}$$

where $F$ is the Faraday 2 form and $J$ the current one-form (i.e. four-vector). The first equation is simply the Bianchi identity, if we think of $F$ as being the exterior derivative of a potential $A$, thus an exact form. In that light, the first equation (which reproduces both Faraday's law and Gauss's law for magnetism) is a purely geometrical one. All the "nasty bits" that arise from spacetime curvature (or some diseased mind's curvillinear co-ordinates on flat space, which can be even nastier) are naturally tamed and concisely summarized in the Hodge star operator, which encodes information about volume forms of various dimension submanifolds relate to one another in spacetime. The Hodge dual's actual calculation is fiddly, though, and in implementation the above is naturally the same as this one here, but the above lets us see what is happenning geometrically.

• What on Earth is $\mathscr Y_0$ supposed to be? – Ryan Unger Jun 27 '17 at 5:13
• @ocelo7 The sign in $\star^2=\pm 1$ most probably – OON Jun 27 '17 at 7:45
• @ocelo7 the freespace wave admittance $\sqrt{\epsilon_0/\mu_0}$ – WetSavannaAnimal Jun 27 '17 at 8:26

A simple way of understanding it is first to note that the Maxwell equations are perfectly compatible with SR (i.e., they are covariant in Minkowski spacetime under Lorentz transformations).

If written in tensorial form they become perfectly compatible with GR (I.e., covariant with respect to any coordinate transformation that is differentiable twice) as long as one replaces the derivatives that appear in Maxwell's equations with covariant derivatives (which include terms with the metric and derivatives of it). Gravity appears, correctly, when that is done.

You can use the EM field a source of the spacetime curvature of you want, and calculate. Those are called electrovac solutions. Or add charged matter also. So it is a very straightforward prescription, just more complex equations.

See a more complicated answer if you want all the details at https://en.m.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime

So it is a very straightforward prescription for turning equations valid in SR, expressed tensorially covariant, into the expected GR equations. This has also been used in taking equations from Quantum Field Theory (ie, applicable to SR transformations), when written covariantly, into Quantum Field Theory in curved spacetimes. One has to take care with any ordering of non-commuting operators. See it at https://en.wikipedia.org/wiki/Quantum_field_theory_in_curved_spacetime