Question: In what type  of frames does light propagate in the same way as it does in special relativity?
About the article "General relativity", section "Relativistic generalization".

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Albert Einstein proposed that the laws of physics should be based on the principle of relativity.
This principle holds that the rules of physics must be the same for all observers, regardless of the frame of reference that is used, and that light propagates at the same speed in all reference frames.
This theory was motivated by Maxwell's equations, which show that electromagnetic waves propagate in a vacuum at the speed of light.
However, Maxwell's equations give no indication of what this speed is relative to.
Prior to Einstein, it was thought that this speed was relative to a fixed medium, called the luminiferous ether.
In contrast, the theory of special relativity postulates that light propagates at the speed of light in all inertial frames, and examines the implications of this postulate.
All attempts to measure any speed relative to this ether failed, which can be seen as a confirmation of Einstein's postulate that light propagates at the same speed in all reference frames.
Special relativity is a formalization of the principle of relativity that does not contain a privileged inertial frame of reference, such as the luminiferous ether or absolute space, from which Einstein inferred that no such frame exists.
Einstein generalized relativity to frames of reference that were non-inertial.
He achieved this by positing the Equivalence Principle, which states that the force felt by an observer in a given gravitational field and that felt by an observer in an accelerating frame of reference are indistinguishable.
This led to the conclusion that the mass of an object warps the geometry of the space-time surrounding it, as described in Einstein's field equations.
In classical physics, an inertial reference frame is one in which an object that experiences no forces does not accelerate.
In general relativity, an inertial frame of reference is one that is following a geodesic of space-time.
An object that moves against a geodesic experiences a force.
An object in free fall does not experience a force, because it is following a geodesic.
An object standing on the earth, however, will experience a force, as it is being held against the geodesic by the surface of the planet.
In light of this, the bucket of water rotating in empty space will experience a force because it rotates with respect to the geodesic.
The water will become concave, not because it is rotating with respect to the distant stars, but because it is rotating with respect to the geodesic.
Einstein partially advocates Mach's principle in that distant stars explain inertia because they provide the gravitational field against which acceleration and inertia occur.
But contrary to Leibniz's account, this warped space-time is as integral a part of an object as are its other defining characteristics, such as volume and mass.
If one holds, contrary to idealist beliefs, that objects exist independently of the mind, it seems that relativistics commits them to also hold that space and temporality have exactly the same type of independent existence. 
As formulated by Albert Einstein in 1905, the theory of special relativity was based on two main postulates:  The principle of relativity — The form of a physical law is the same in any inertial frame.
The speed of light is constant — In all inertial frames, the speed of light c is the same whether the light is emitted from a source at rest or in motion.
(Note this does not apply in non-inertial frames, indeed between accelerating frames the speed of light cannot be constant.
Although it can be applied in non-inertial frames if an observer is confined to making local measurements.)
There have been various alternative formulations of special relativity over the years.
Some of these formulations are equivalent to the original formulation whereas others result in modifications. 
Theories not equivalent to special relativity.
Test theories.
A number of theories have been developed to allow assessment of the degree to which experimental results differ from the predictions of relativity.
These are known as test theories and include the Robertson and Mansouri-Sexl (RMS) theories.
This test theory uses Einstein synchronization in a "preferred" frame, while all other frames take over the value of this "preferred" frame by "external synchronization".
However, this alone does not provide any testable deviation from special relativity, therefore they included additional parameters, making the two-way speed of light anisotropic in this model.
To date, all experimental results agree with special relativity within the experimental uncertainty.
Another test theory is the Standard-Model Extension (SME).
It employs a broad variety of coefficients indicating Lorentz symmetry violations in special relativity, general relativity, and the Standard Model.
Some of those parameters indicate anisotropies of the two-way and one-way speed of light.
However, it was pointed out that such variations in the speed of light can be removed by suitable redefinitions of the coordinates and fields employed.
Though this doesn't remove Lorentz violations "per se", it only shifts their appearance from the photon sector into the matter sector of SME (see above Experiments on two-way and one-way speeds using the Standard-Model Extension. 
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.
In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics.
(The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.)
The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with high-energy phenomena.
With Lorentz symmetry, additional structures come into play.
They are defined by the set of light cones (see the image on the left).
The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image).
These sets are observer-independent.
In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor.
In mathematical terms, this defines a conformal structure.
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected.
Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames.
Instead there are approximate inertial frames moving alongside freely falling particles.
Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames.
But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf.
below).
The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.
The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity.
In the language of spacetime geometry, it is not measured by the Minkowski metric.
As in the Newtonian case, this is suggestive of a more general geometry.
At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian.
Consequently, we are now dealing with a curved generalization of Minkowski space.
The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric.
Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).
Einstein developed special relativity on the basis of two postulates: First postulate – Special principle of relativity – The laws of physics are the same in all inertial frames of reference.
In other words, there are no privileged inertial frames of reference.
Second postulate – Invariance of "c" – The speed of light in a vacuum is independent of the motion of the light source.
Special relativity can be derived from these postulates, as was done by Einstein in 1905.
Einstein's postulates are still applicable in the modern theory but the origin of the postulates is more explicit.
It was shown above how the existence of a universally constant velocity (the speed of light) is a consequence of modeling the universe as a particular four-dimensional space having certain specific properties.
The principle of relativity is a result of Minkowski structure being preserved under Lorentz transformations, which are "postulated" to be the physical transformations of inertial reference frames.