It doesn't make sense to apply Lorentz transformations to the Schwarzschild-or any, curved-metric. Lorentz transformations, by definition, are those transformations that leave the flat-Minkowski-metric invariant. They don't and can't leave a curved metric invariant.

@Stam Nicolis

Behind the Lorentz transformations, as well as the Schwarzschild metric, there are real or at least observable distortions of length and time. If a particle or body moves in a gravitational field, these distortions will produce some general result. The fact that applying the Lorentz transformations to a curved metric violates invariance does not mean that the result will be erroneous within certain limits. For example, if the curvature of space-time due to the gravitational field is small compared to the distortion of length and time caused by the presence of the Lorentz factor, then in this limit they will most likely give a result close to the observed one. The equations of Newton's law of gravitation are not invariant with respect to the Galilean transformations, but this does not mean that they are incorrect within certain limits. The fact that in the general case the application of the Lorentz transformations leads to non-invariance is a shortcoming of the theory, but this does not mean that it is not applicable.

It doesn't make sense to apply Lorentz transformations, precisely because they aren't isometries of the Schwarzschild metric. The transformations that it does make sense to apply are general coordinate transformations, these are the isometries of general relativity.

Indeed, the Schwarzschild metric is, already, the result of applying a subset of these transformations: It's the unique static metric, solution to Einstein's equations, that is asymptotically flat at infinity and depends on only one parameter, which can be identified with the mass of the body in question. It is indeed possible to show that the assumption that the metric is static implies that it is spherically symmetric, also.

Einstein's equations are not invariant under global Lorentz transformations, but under general coordinate transformations. And vice versa, the equations for the metric, that are invariant under general coordinate transformations, are Einstein's equations.

Newton's equations, for any potential, that isn't explicitly time-dependent, are invariant under Galilean transformations, that's why these are relevant for Newtonian mechanics.

The symmetries come first and determine the dynamics. Imposing invariance under global Lorentz transformations is equivalent to imposing that the spacetime geometry is flat (and vice versa).

You cannot go from the Schwarzschild metric to the Kerr metric using isometries or any other coordinate transformations, although they describe the gravitational field of the same spherical body. But a physically rotating and static body are different situations.

And in the case under consideration, the resulting metric describes a non-stationary spherical source of gravity. Although the use of Lorentz transformations assumes that space-time is close to flat, that is, weak gravity.

The question

“Is it possible to apply Lorentz transformations to the Schwarzschild metric?”

- on one hand has rather evident answer – not; Lorentz transformations are completely and really effectively applicable only in flat spacetimes – in both, 4D Euclidian one with metrics (ct,X,Y,Z), as that was/is in Lorentz-Poincaré theory, and in rather strange imaginary Minkowski 4D space with metrics (icτ,X,Y,Z) in the SR,

- but from other hand really the question hasn’t real practical sense – as that is rigorously scientifically proven in the Shevchenko-Tokarevsky’s Planck scale informational physical model, in this case see first of all

https://www.researchgate.net/publication/383127718_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics , section 6.“Mediation of the fundamental forces in complex systems” , [though, of course, to understand why presented in this section initial models of fundamental Nature Gravity, Electric, and Nuclear/Strong forces really are practically for sure scientifically correct it is necessary to read whole paper]

- Matter’s spacetime is fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), where really space cτ-dimension in the Euclidean and Minkowski metricses above is postulated as the time ct-dimension.

At that the point above isn’t too essential in everyday physical practice, since at not too large energy/momentum interactions [when no having rest mass particles are created] the fundamental Forces above act only in 3D space, and so, including, Lorentz transformations are well applicable in the real utmost universal [5]4D Cartesian spacetime with metrics (ct,X,Y,Z,ct) and in the conventional metricses above.

Including after the initial Gravity model above will be developed into a comprehensive enough theory, action of Gravity will be described/analyzed including according to the transformations; - quite equally as that is now in classical electrodynamics; that is another thing that it should be corrected in accordance with the initial Forces models approach above.

Though see also a couple examples where using of metrics (cτ,X,Y,Z,ct) instead of metrics (ct,X,Y,Z) is principal see in SS posts in https://www.researchgate.net/post/What_is_the_actual_physical_meaning_of_the_Lorentz_Transformations#view=671d0eaa1a9322f7030e5a2d/307/307/308/308 , 2 posts, page 309, and

https://www.researchgate.net/post/Is_there_a_reasonable_alternative_to_the_theory_of_the_expanding_universe/1054 two posts, page 1054

Cheers

on one hand has rather evident answer – not; Lorentz transformations are completely and really effectively applicable only in flat spacetimes – in both, 4D Euclidian one with metrics (ct,X,Y,Z),

Analysis of the frequency of the Pioneer 10 signal shows that the application of the Lorentz transformations to the Schwarzschild metric is possible. This is because the space is approximately flat.

This

“…Analysis of the frequency of the Pioneer 10 signal shows that the application of the Lorentz transformations to the Schwarzschild metric is possible. This is because the space is approximately flat.…..”

- again is rather speculative claim. Really – see SS post above – Matter’s spacetime fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct),

- and so application in physics of any other spacetimes really is senseless, including can result – and really results – only in some irrational illusory description and analysis of what really exists and happens in Matter.

Just since all real physics works in flat, even 4D [while utmost universal “kinematical” spacetime is [5]4D Cartesian spacetime with metrics (ct,X,Y,Z,ct)], and, moreover, even in that has strange imaginary mathematically either space or time, Minkowski space,

- where all besides Gravity fundamental Nature Weak, Electric, and Strong, forces act,

- the physics in most – though principally not in fundamental - cases, is adequate to the reality.

Gravity is fundamentally is nothing else than “the forth” fundamentally Nature force, which acts in the flat spacetime with [as really all other Forces act] at least [there can be, in principle some other dimensions] the [4+4+1] 4D metrics above.

More see the SS post above and links in the post, to read SS posts in https://www.researchgate.net/post/NO47What_is_Force_a_Field_Where_is_the_Force_Field_How_does_it_appear_Is_the_Force_Field_a_Regulating_Effect_of_the_Energy-Momentum_Field/2 about what are Gravity, Electric, and Nuclear/Strong Forces , pages 1,2

- ir would be well useful also.

reDzennn comment [5-1 passages, the passage with links to SS&VT papers is removed] in https://phys.org/news/2024-12-nature-constant.html is rather relevant to this thread question.

Cheers