Grinding through the algebraic analysis of the Einstein-Maxwell field equations for gravity and electrodynamics is laborious and easy to get wrong. So I'll try to write a tool here that'll help automate that and, with any luck, be less apt to get it wrong. I want to be able to do this in a way that yields the (X)HTML I need when writing up a derivation on my web site: so it has to not only do the grinding, but also stop at intervals along the way so that I can have it emit a sequence of stages in the derivation in which a human reader has some chance of seeing how we get from each point to the next. I'll be happy if I can initially just display the answers: later, I'll worry about stopping at handy points along the way.
The work is at least all fairly regular:
so I'll try to make the page follow roughly this work-flow. I'll allow
you to type * for the tensor product operator, which I'll convert
to × (you can write × if you like) and I'll offer to let you use
an alias for any Greek-letter symbol you're using, to save typing. Here are
some names you aren't at liberty to change from their fixed meanings:
the covariant differential operator
the intrinsic differential operator: this maps any scalar field h to its gradient dh; and any alternating form w to its alternating derivative d^w
the metric of space-time
the electromagnetic field tensor
the trace-permutation operator on tensors
the list whose four members are the gradients of your chart; this is the basis of co-vectors
the list whose four members are the vectors forming the basis dual to b; so that b(i)·p(j) = δ(i,j), i.e. it's 0 unless i = j, in which case it's 1.
the speed of light
Newton's gravitational constant
the impedance of free space
the transcendental function defined by the power series exp = (: sum(: power(i, x)/factorial(i) ←i :{naturals}) ←x :)
the transcendental functions which map a scalar, a, to the (scalar) sine and cosine of the angle a.radian; these satisfy exp(i.z) = cos(z) +i.sin(z) when i is a square root of minus one
the result of dividing a Euclidean circle's circumference by its diameter; we could, equally, specify it as half the period of sin and cos.
(at least for the present: letting you change them won't be a high priority until the rest works !).
Note: this page uses the repeating form controls features of the Web Forms 2 spec. Various buttons marked ↓ won't work for you unless your user agent (a.k.a. browser) supports that. I confess to being guilty of laziness – I could write some JavaScript/DOM code to achieve the same effect for older browsers, but the browser I'm actually using, Opera, does support this feature, so that compatibility work is low priority, at least until this page is fully working and someone who wants to use it asks for compatibility !
OK, that lets me infer a basis, b, of covectors by applying d to each of the scalar fields you've named as your chart members. There is then a dual basis of vectors implied by this; (only) for simple enough charts, this dual basis's member matching the derivative of a given co-ordinate is in the direction along which that co-ordinate increases most rapidly – for example, a unit time-like vector is commonly the dual's member corresponding to the gradient of the time co-ordinate. I'm going to need to refer to the members of this dual basis, so you might want to give them names (otherwise, I'll just refer to them as p(0), p(1), p(2) and p(3) when I need to).
You can also identify some expressions as important, optionally giving them names, and provide short-forms for some variables (e.g. ones with Greek symbols for their names). I'll recognize the latter in your typed input but substitute their proper names when displaying the results. When I'm more sophisticated, I'll try to exploit the expressions you tell me about as common-subexpressions to track in my analysis; and I'll substitute any name you give for them when I notice that I can.
You need to specify the metric of space-time, g, and the electromagnetic field tensor, F. Each is a tensor of rank covector&tensor;covector and F is antisymmetric, so you may find * or × useful to denote tensor multiplication and ^ to denote its anti-symmetric variant, a^b = (a×b −b×a)/2.
I failed to make sense of what you typed here: with any luck the following notes shall help you work out why and adjust to something my simplistic parser can handle:
That's not going to work. When I try to invert the metric, it turns out to be singular. Did you mis-type something ?
The electromagnetic field should be an antisymmetric tensor: what you've given has symmetric part
Did you mis-type something ?
OK, that's all the input you need to give. The metric's inverse is
The metric's non-zero components have derivatives:
The 4-current, j, is a vector field satisfying a conservation law, sum(: p(i)·d(j·b(i)) ←i :) = 0. I instantly recognize this as a sign that j is the result of dividing, by the measure, an alternating form of rank one less than the dimension of space-time: j's conservation law then says that d^ annihilates this form. The measure, μ, is a square root of (a scalar multiple of) the determinant of the metric, μ×μ = −det(g); it mediates integration on our smooth manifold (it's also the totally antisymmetric form Ε in Einstein's notation). So d^(μ·j) = 0 is our conservation law. (In any topologically trivial domain, this implies that μ·j is d^ of some alternating tensor of rank dim−2; the field equations give us this tensor.)
We can now grind our way through Maxwell's equations, which don't actually need the covariant derivative.
Hereafter, it is convenient to replace F with
which has the same dimensions as the metric times the square of the speed of light.
The co-variant derivative, D, – the differential operator that the physics believes in – is characterized by the fact that it deems the metric, g, constant: D(g) = 0. From this it's possible to derive a (basis-dependent) tensor which, when contracted with a member of our basis of co-vectors, yields the derivative of that basis member. That tensor is:
From this we can compute the derivatives of individual members of our basis, b, of covectors:
From the covariant derivative, we can compute Riemann's tensor:
A trace of this, the Ricci tensor, encodes the curvature of space-time:
The energy-momentum-stress content of the electrodynamic field is:
Combining this with the curvature and metric, we can infer the energy-momentum-stress content of whatever matter may be present:
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Written by Eddy.