I am not sure what your comment has to do with the recent images of a black hole. And carefully, what do you mean that Einstein's field equations do not work on a torus? (I am not disagreeing or agreeing, before I can do that I need to know what you mean;)

dear kind mind Andrew James Bruce yes you are right, black hole event horizon is observed and this confirms einstein relativity, but at the same time a torus object surrounding a black hole is observed and einstein field equastions do not work on torus geometry or torus metric, then naturally we may ask the validity or limits of einstein relativity?

Ayhan, why do you say an event horizon was observed when an event horizon would be, by its very definition, unobservable?

It is not Einstein's theory of general relativity that gives rise to the common belief in black holes, but the academic community's misunderstanding of Einstein's theory. When he formulated his theory he thought the universe had existed forever. Some years later, as the big bang model gained traction, he realised that the finite age of the universe completely precludes the formation of black holes and event horizons. He published this conclusion on September 1st 1939, the same day that Oppenheimer & Snyder published an article in which they misinterpreted the results of their own calculations - and World War II broke out.

Since event horizons cannot form, there is nothing to prohibit the collapsed objects found at the centre of each galaxy from assuming a toroidal geometry. Indeed, the existence of highly energetic quasars and other active galactic nuclei rather confirms the expectations of general relativity and shows that Einstein was right to maintain that gravitational time dilation prevents collapse from proceeding all the way to black hole formation. The toroidal structure of the central engines is betrayed by the electrically charged outflows, magnetised jets moving at ultrarelativistic velocities: spheroidal black holes should be electrically neutral and incapable of accelerating charged particles:

Preprint Dispelling Black Hole Pathologies Through Theory and Observation

Article Quasars: A supermassive rotating toroidal black hole interpretation

Robin Spivey dear kind mind friend, when you or one derive einstein field equations on torus geometry it is naturally to see this, einstein field equations do not work on torus metric, but despite all for ages we observe saturn or jupiter rings, always I may imagine torus objects in universe, for that conclusion it is not needed to study hundreds pages of physics or math,

Ayhan, the articles I linked to cite numerous works involving solutions to the field equations with a toroidal geometry. Since toroidal objects can obviously arise in Newtonian gravity, the same is also true of general relativity. Basic intuition suffices to appreciate that this is true, without recourse to mathematics.

just like schwartzian metric, it is easy to derive metric on torus, and following same steps, einstein field equations may be derived for torus metric, and for 0th and 1 st equations a solution maybe derived very similar to schwartzian solution, but finally 2nd and 3 rd equations are not satisfied. my question is not toroidal objects indeed it is torus geometry or torus metric,

Dear Ayhan,

Einstein's equations can have a metric with torus symmetry, for instance a selfdual Weyl curvature and two linearly independent commuting Killing fields.

But in the black hole the torus is due to have an accreation gas distribution of matter outside of its event horizon. This can be determined even using a Schwarzchild metric provided the mass, angular momentum and charge of the black hole can be known. The black hole is just a singularity of the gravitational field which physically corresponds to a very high density of mass and its surrounds can be considered within its field of gravitation.

few days later here I will share my notes handwritten by me, original copies, how I derived torus metric and then einstein field equations, but first two einstein field equations a solution is possible, just like schwartzschild solution it has two singular points, the solution may collapse into black ring, but 3rd and 4th einstein field equations do not satisfy it, this is nearly semi einstein solution,

my calculations on einstein field equations on torus, more than 10 years ago I did these calculations,