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Pathological coordinates and special coordinates

If a time barrier prevents formation of black holes and if anything that approaches a very compact mass experiences instant evaporation, how can some still claim black holes form? One fallacy that has contributed to the mistaken belief that black holes form in Einstein's relativistic space time is the labelling of ordinary coordinates as "pathological" and the assertion  that there exist "special" coordinates that can be used to show a mass will compact to form a black hole.

Coordinates are reference systems that make measurements from a particular frame of reference. The “special” coordinates utilize a reference frame that travels along with a traveler, either at the same velocity or faster, making it appear that black holes can form. However, such use of a “co-moving” reference frame to show black holes can form employs a logical fallacy called begging the question, where the proposition to be proved is assumed to be true in a premise.

Let’s see if switching coordinate’s can help break through the time barrier. Remember, in Bob’s adventure he left Betty a distance away and then traveled down to the very compact mass on his own. He evaporated before reaching the surface. To Betty it appeared that the evaporation took a very long time. To Bob, it felt like the evaporation happened instantly. This is because at the time of evaporation they were in different reference frames that,  because of the effect of gravity, measure time at different rates.

Betty saw what happened to Bob, but was still curious about what it was like down on the surface of the compact mass. She consulted with her friend Barb, a theoretical physicist. Barb explained that Bob’s experience of instant evaporation resulted because his journey was measured from Betty’s reference frame a long distance away from the compact mass.  In Betty's reference frame, it took Bob a very long time to complete the journey. If only Barb had traveled down with Bob. Then, according to measurements from Barb's reference frame, Bob could have reached the compact surface in a very short amount of time, allowing Bob to easily complete the journey before evaporating.

Is Barb right? Does measuring from a different reference frame achieve a different result? Of course not. This is a classic case of begging the question. The proposition to be proved is whether Bob (or anyone) can reach the surface of the compact mass without evaporating. Barb tries to prove this by measurements made from her reference frame. The implicit premise is that Barb, traveling with Bob, survives long enough to measure the time it takes Bob to arrive at the surface of the compact mass, which requires that Barb herself reach the surface with Bob. The proposition (someone can reach the surface) has been assumed to be true in the premise (Barb can reach the surface). The question has been begged.

While Barb’s logic may seem a bit far fetched, this is actually one explanation that has been used to contradict Albert Einstein's assertion that black holes cannot form. Ordinary coordinates (i.e., measurement's from Betty's reference frame) that show black holes cannot form are labeled "pathological". Only special "co-moving" coordinates such as Novikov coordinates, Kruskal-Szekeres coordinates, or Eddington-Finkelstein coordinates (i.e. measurements from Barb's reference frame) are allowed to measure the progress of a collapsing surface forming a black hole. Novikov coordinates exactly co-move with with the surface forming the black hole. Ingoing Eddington-Finkelstein coordinates move a bit faster since they utilize an inbound photon as the reference frame. But the logical fallacy is the same.

This artifice used to contradict Albert Einstein's view black holes do not form was proposed only after his death. It is inconceivable this logical fallacy would not have been immediately uncovered were he alive to refute it.

For a more rigorous consideration of the subject matter see Five fallacies used to link black holes to Einstein’s relativistic space-time

Copyrighted Article--by Doug Weller--used with permission

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