explaining gravity: from The Goon Show

Relativity and Gravitation

My pages (generally fragmentary):

General Relativity

General Relativity is Einstein's (classical) theory of gravitation and the geometry of the universe on large scales. The beauty of the theory is that it is obtained by careful reasoning from some very simple grounds:

Locally Inertial Frames
exist: to put it another way, there are frames of reference with respect to which, within some (possibly small) domain, the momentum of any isolated small system (small on the scale of the domain) within the domain does not vary (or, rather, varies only to a degree small in relation to the system's size, relative to the domain's scale, and to the distance of the system from the centre of the domain).
Space is Locally Euclidean.
For any point in the universe, there is some locally inertial frame whose domain contains a neighbourhood of that point which is locally Euclidean: which means that it can be mapped by a chart using a portion of an actual Euclidean space to represent the neighbourhood of the point. The dimension of the Euclidean space involved is the same for all points of the universe. This leads to the conclusion that geodesics are the trajectories along which mass and energy flow when subject to no other force than gravitational (i.e. mass-interaction).
The Speed of Light
is the same, regardless of the source of the light or the direction in which it is propagating, in all locally inertial frames. This leads to the conclusion that the natural metric of space-time, which defines the geodesics above, is not positive-definite: all light-like geodesics have zero length (proper time) with respect to this metric. This leads to a partition of the collection of geodesics into: space-like (imaginary proper time), forward and backward time-like (real proper time: positive if forward, negative backward) and light-like. The latter form the boundary between time-like and space-like domains and can be subdivided into forward and backward by consideration of which time-like domain they border.
Make it simpler
Mass/Energy
Inertial and Gravitational mass are the same thing. (The theory ends up inferring that they're also the same thing as energy.)

Inevitably, there's more to it than that but this is what I can remember off the top of my head as an aside while writing about quantum mechanics.

Reasoning from such simple premises leads ultimately to Einstein's field equations for general relativity, which relate the energy-momentum-stress tensor, T, (which describes the presence of matter) to the Ricci tensor, R, (which describes the curvature of space-time) according to:

wherein κ is Einstein's gravitational constant (equal to 8.π.G (times a suitable power of the speed of light), where G is Newton's gravitational constant), Λ is the Cosmological Constant, g is the metric of space-time and g\R (pronounced g under R by analogy with R over g for R/g) is the result of contracting g's inverse on the left of R, a.k.a. ig·R if ig were g's inverse. Note that Λ appears in such a rôle that it may be treated as though it were (half) the (negative) diagonal entry of g\R in an extra dimension.

Note, correspondingly, the equation connects T's components in our macroscopic dimensions to R's (small) components in these dimensions combined with a term in the metric scaled by R's trace – which may contain large terms due to any microscopic dimensions of space-time. Consequently, analysis of macroscopic space (which only tells us the (small) portion of g\R's trace due thereto) may be expected to give a radically different value of Λ from analyses influenced by the microscopic dimensions (e.g. quantum mechanical analyses based on the background energy of free space): the difference is exactly the contributuon to g\R's trace due to any microscopic dimensions. [This all presumes a widely-expressed view that space-time has four slightly curved dimensions and some other dimensions, to make up a total of about 10 or about 26, which are tightly curved, so that we never notice them.]

T contains a contribution from the electromagnetic field, which I discuss elsewhere. This is quadratic in the electromagnetic field tensor, which encodes the electric and magnetic fields. In general, it is supposed that T satisfies τ[*,0,*](DT/g) = 0, which I should check for the electromagnetic contribution.

The weak field limit

The metric, g, includes a c.dt×c.dt.exp(2.φ/c/c) term with φ being the gravitational potential, φ = −G.M/r in the Newtonian approximation with M being the mass of the system. Now, where did I derive that in my notes…

See also

Wikipedia has plenty of exact solutions to the general relativistic field equations, collected as a category.

Stray notes

Autumn 1997, minor detail … Jeremy tells me Yang-Mills takes a general Lie group (or its Lie algebra) and produces chromodynamics without the quantisation on a smooth manifold, along with an account of the non-linearities that make up the boson-boson interactions. Yumm ;^)

Late summer 2007, detail: a technical explanation of technical explanation tells me, in passing, that the perihelion advance of Mercury predicted by Newtonian mechanics, using the best Victorian data, was 5557 seconds of arc per century; and the measured advance was 5600 seconds of arc per century. General relativity accounts for the other 43 seconds of arc per century. I am impressed that the Victorians had the computational power to determine their prediction, the astronomical expertise to measure the result, the confidence in the precision of each to be aware of the tiny difference and the intellectual integrity to acknowledge that it presented a problem for the theory.


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