<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article  PUBLIC "-//NLM//DTD Journal Publishing DTD v3.0 20080202//EN" "http://dtd.nlm.nih.gov/publishing/3.0/journalpublishing3.dtd"><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="3.0" xml:lang="en" article-type="research article"><front><journal-meta><journal-id journal-id-type="publisher-id">IJAA</journal-id><journal-title-group><journal-title>International Journal of Astronomy and Astrophysics</journal-title></journal-title-group><issn pub-type="epub">2161-4717</issn><publisher><publisher-name>Scientific Research Publishing</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.4236/ijaa.2012.24031</article-id><article-id pub-id-type="publisher-id">IJAA-26225</article-id><article-categories><subj-group subj-group-type="heading"><subject>Articles</subject></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Physics&amp;Mathematics</subject></subj-group></article-categories><title-group><article-title>
 
 
  Kruskal Coordinates and Mass of Schwarzschild Black Holes: No Finite Mass Black Hole at All
 
</article-title></title-group><contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>bhas</surname><given-names>Mitra</given-names></name><xref ref-type="aff" rid="aff1"><sub>1</sub></xref><xref ref-type="corresp" rid="cor1"><sup>*</sup></xref></contrib></contrib-group><aff id="aff1"><label>1</label><addr-line>Theoretical Astrophysics Section, Bhabha Atomic Research Centre, India</addr-line></aff><author-notes><corresp id="cor1">* E-mail:<email>amitra@barc.gov.in</email></corresp></author-notes><pub-date pub-type="epub"><day>31</day><month>12</month><year>2012</year></pub-date><volume>02</volume><issue>04</issue><fpage>236</fpage><lpage>248</lpage><history><date date-type="received"><day>September</day>	<month>6,</month>	<year>2012</year></date><date date-type="rev-recd"><day>October</day>	<month>7,</month>	<year>2012</year>	</date><date date-type="accepted"><day>October</day>	<month>18,</month>	<year>2012</year></date></history><permissions><copyright-statement>&#169; Copyright  2014 by authors and Scientific Research Publishing Inc. </copyright-statement><copyright-year>2014</copyright-year><license><license-p>This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/</license-p></license></permissions><abstract><p><html>
 <head></head>
 
  When one presumes that the gravitational mass of a neutral massenpunkt  is finite, the Schwarzschild coordinates <img style="width:31px;height:19px;" alt="" src="Edit_167d5248-d381-412d-829e-2cfa960a07c6.bmp" width="41" height="14" />
  appear to fail to describe the region within the event horizon (EH), <img style="width:43px;height:18px;" alt="" src="Edit_f0025204-44cb-43eb-8bac-d06489d5c607.bmp" width="60" height="21" /> of a Schwarzschild Black Hole (SBH). Accordingly, the Kruskal coordinates <img style="width:27px;height:17px;" alt="" src="Edit_395697d5-dce0-4b94-8be9-1d1aa24ae7ec.bmp" width="35" height="21" />
   
  were invented to map the entire spacetime associated with the SBH. But it turns out that <img style="width:64px;height:14px;" alt="" src="Edit_f6ca4b38-8b25-4150-a1f0-5c382f2f5b8f.bmp" width="83" height="10" /> at the EH (Mitra, IJAA, 2012), and the radial timelike geodesic of a point particle would become null. Physically this would mean that, the EH is the true singularity, i.e., <em>M </em>= 0, and this zero mass BH could only be a limiting static solution which must never be exactly realized. However, since in certain cases <img style="width:79px;height:12px;" alt="" src="Edit_9d715115-2bfd-48a5-94a1-0669e0267cee.bmp" width="92" height="12" />, here we evaluate this derivative in such cases, and find that, for self-consistency, one again must have <img style="width:63px;height:12px;" alt="" src="Edit_f6ca4b38-8b25-4150-a1f0-5c382f2f5b8f.bmp" width="83" height="10" /> at the EH. This entire result gets clarified by noting that the integration constant appearing in the vacuum Schwarzschild solution (and not for a finite object like the Sun or a planet), is zero (Mitra, J. Math. Phys., 2009). Thus though the Schwarzschild solution for a point mass is formally correct even for a massenpunkt, such a point mass or a BH cannot be formed by physical gravitational collapse. Instead, physical gravitational collapse may result in finite hot quasistatic objects asymptotically approaching this ideal mathematical limit (Mitra &amp; Glendenning, MNRAS Lett. 2010). Indeed 
  “
  the discussion of physical behavior of black holes
  , classical or quantum, is only of academic interest
  ”
   (Narlikar &amp; Padmanbhan, Found. Phys. 1989).
 
</html></p></abstract><kwd-group><kwd>Kruskal Coordinates; Black Hole; Black Hole Alternatives; Eternally Collapsing Object</kwd></kwd-group></article-meta></front><body><sec id="s1"><title>1. Introduction</title><p>The concept of Black Holes (BHs) is one of the most important plinths of modern physics and astrophysics. As is well known, the basic concept of BHs originally arose due to Laplace, more than 200 years ago, in the cradle of Newtonian gravitation, where the mass of a body remains fixed during collapse. In General Theory of Relativity (GTR), the gravitational mass is less than the baryonic mass<img src="8-4500105\9da611e9-f4c2-46c1-af70-3200c68bdf8f.jpg" />. Further, as the body contracts and emits radiation <img src="8-4500105\806181a3-d1de-49dc-8a46-391d52f09d4c.jpg" /> keeps on decreasing progressively along with <img src="8-4500105\faf5623b-6a82-4359-85b2-2cabc5f12674.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref1">1</xref>]. Thus, given an initial gravitational mass<img src="8-4500105\51fb9fec-a394-4fdb-bb0e-fddb5745501b.jpg" />, one can not predict with certainty the value of <img src="8-4500105\a7314a06-b790-45a5-8fe7-786326631dee.jpg" /> when we would have<img src="8-4500105\8f76bf78-831f-422c-8398-38265adf9cc7.jpg" />. Neither are the values of<img src="8-4500105\26eac07a-6273-44c5-bc0b-420b966f006c.jpg" />, <img src="8-4500105\e17cd0af-3170-4047-b1d6-f2413b2ceece.jpg" />and <img src="8-4500105\128bdfff-49ca-44a8-9956-5e39cf636014.jpg" /> related by any combination of fundamental constants though, it is generally assumed that<img src="8-4500105\bd3b2d4b-a7d4-4540-8676-2b75fce3f59a.jpg" />. Ideally, one should solve the Einstein equations analytically to fix the value of <img src="8-4500105\873e5b49-9fba-4272-a15b-c35e87c1a5d6.jpg" /> for a given initial values of <img src="8-4500105\898b7949-7f22-49ba-9ce0-d0150280592b.jpg" /> and <img src="8-4500105\4d07f1e9-045b-4065-99d7-5dbb02b0d81d.jpg" /> for a realistic equation of state (EOS) and energy transport properties [2,3]. However even when one does away with the EOS by assuming the matter to behave like a dust, <img src="8-4500105\b145ef6b-6670-48d6-a5da-789227b9c6d1.jpg" />, one does not obtain any unique solution if the dust is inhomogeneous. Depending on the various initial conditions and assumptions (like self-similarity) employed one may end up finding either a BH or a “naked singularity” [<xref ref-type="bibr" rid="scirp.26225-ref4">4</xref>]. By further assuming the dust to be homogeneous Oppenheimer and Snyder (OS) [<xref ref-type="bibr" rid="scirp.26225-ref5">5</xref>] found an asymptotic solution of the problem by approximating Equation (36) of their paper [2,3]. The region exterior to the event horizon <img src="8-4500105\0b467897-d633-4d4a-bfbf-d75bd269ed08.jpg" />can be described by the Schwarzschild coordinates <img src="8-4500105\1f42e9dc-372e-43b2-8cee-6dc607577e52.jpg" /> and <img src="8-4500105\73fa8d40-5541-4b8f-981f-5822954c3ab2.jpg" /> [6,7]:</p><disp-formula id="scirp.26225-formula143285"><label>(1)</label><graphic position="anchor" xlink:href="8-4500105\41ca3374-8a5b-4c84-9af9-1282be2fefae.jpg"  xlink:type="simple"/></disp-formula><p>where<img src="8-4500105\ebb3f006-a5a6-439e-8dbf-974de68dfc8d.jpg" />, <img src="8-4500105\925fe674-45a1-41f9-9441-92d71735c6f5.jpg" />, <img src="8-4500105\964d314c-cb5d-453e-9b14-a434d5d59c1e.jpg" />, and<img src="8-4500105\492cadf2-4e4e-4a66-a040-4f312db7f567.jpg" />. Here, we are working with a spacetime signature of <img src="8-4500105\1847c4a6-538b-495b-8f37-9decbe96549c.jpg" /> and <img src="8-4500105\eb3b545b-61b9-477e-a4e3-0b1885b58462.jpg" /> has a distinct physical significance as the invariant circumference radius. For<img src="8-4500105\c7b51250-9f8b-4776-b71c-1baf9eb9c680.jpg" />, the worldline of a free falling radial material particle is indeed timelike <img src="8-4500105\9d59ddf7-02fb-47ae-a107-be4fb99d1add.jpg" /> and the metric coefficients have the right signature,</p><p><img src="8-4500105\193cf55e-adb6-442f-bab2-dfde61386cb9.jpg" />But at <img src="8-4500105\582f28c1-5701-46f0-b821-e54416fc4ec4.jpg" /> <img src="8-4500105\3997ee68-2064-4fa6-99dc-2c0994e9d4b3.jpg" /> blows up and as<img src="8-4500105\3e0a607c-5d20-481d-8c10-722c1b9afa2f.jpg" />, the <img src="8-4500105\d533904f-8124-486d-8ed3-bbb31bcd09eb.jpg" /> and <img src="8-4500105\47d0513e-fc31-48fc-9e9a-9006c8c840a4.jpg" /> suddenly exchange their signatures though the signatures of <img src="8-4500105\74df6a90-5dbc-4652-82af-9baf48c75f45.jpg" /> and <img src="8-4500105\0c7fe4c2-b7d8-49ba-843a-40ae1a036f98.jpg" /> remain unchanged. This is interpreted by saying that, inside the event horizon, <img src="8-4500105\90459e70-144f-4aec-8888-40ac2a25c66b.jpg" />becomes “time like” and <img src="8-4500105\b8b3f00d-92ad-4382-806d-6a45b4da8333.jpg" /> becomes “spacelike”. However, we see that actually <img src="8-4500105\ca4550cd-7cd9-4cd8-996f-c57927de28d6.jpg" /> continues to retain, atleast partially, its spacelike character by continuing to be the “invariant circumference radius”. Also, note that, if physically meaningful quantities like the Rimennian curvature components behaved like <img src="8-4500105\67d78796-1c75-47f1-afc7-3d9b9ca58c1d.jpg" /> outside the EH, they continue to behave in a similar manner, and not like <img src="8-4500105\35c84ad6-a3c7-44a8-83f8-146bcbada3b4.jpg" /> inside the EH. And it should be borne in mind here that by a fresh relabelling or by any other means, the curvature components can not be made to assume the form<img src="8-4500105\141e5912-03b7-4c1b-bc29-f392feb1f02a.jpg" />. One particular reason for this is that, we would see later that, inside the EH, we have <img src="8-4500105\724209c1-0ad9-4702-aaef-ec3a4cd86e98.jpg" /> while, of course, the value of <img src="8-4500105\7f3e649b-4b4a-439f-bfa2-9952bacd9956.jpg" /> remains finite. Thus it may not actually be justified to conclude that <img src="8-4500105\42cc90f0-e25c-4e39-b09e-96f329d735db.jpg" /> becomes the “timelike coordinate” inside the EH even though <img src="8-4500105\6f355d41-3ad0-4182-87a7-1615b0f589ba.jpg" /> changes its sign.</p><p>So far, it has not been possible to resolve this enigma of the duality in the behaviour of <img src="8-4500105\7152c6e8-18ed-4256-aea6-e92ace5815e4.jpg" /> for<img src="8-4500105\5b7d6e97-7592-4e24-8bd1-a61fcdb947d0.jpg" />, and the present paper intends to attend to this problem. Since Kruskal coordinates <img src="8-4500105\cd3e79ec-468c-4a24-8152-208d27faf9d3.jpg" /> are believed to properly chart the SBH spacetime, it is imperative that, one studies the properties of these coordinates. In a recent paper (Pap. I) we found that the Kruskal derivative <img src="8-4500105\4d36d548-d1f6-4c1e-96c1-7682c6744c20.jpg" /> at the EH [<xref ref-type="bibr" rid="scirp.26225-ref8">8</xref>]. However, this derivative can be obtained from various directions, and in some cases, <img src="8-4500105\905bcc73-18b9-4a44-b9d0-b5a484bc65d7.jpg" />may assume a <img src="8-4500105\c739c8bd-0be2-4968-9bdd-a70ca7bd2e72.jpg" /> form as<img src="8-4500105\6b71d68b-eef7-4860-bb2f-b5f193cffd3d.jpg" />. Here we would like to closely examine such cases. Eventually, it would be found that, for self-consistency, one must have <img src="8-4500105\6f9f7eeb-8de0-431b-80df-62aadbf8bc1c.jpg" /> at<img src="8-4500105\2dc52300-6938-439a-81e9-eda735a22e6c.jpg" />. This result would suggest, that for a neutral point mass, the integration constant appearing the SBH solution is<img src="8-4500105\afb42bc0-ce68-476b-a8c6-ce1aa19f62cb.jpg" />. Then, as we will soon see, the Kruskal coordinates implicitly involve division by zero, and which explains various oddities associated with them. We would also take note of the objections associated with the Kruskal coordinates by some other authors.</p></sec><sec id="s2"><title>2. Kruskal Coordinates</title><p>Although <img src="8-4500105\44d74ca8-9f81-4565-9227-9e1cd1b11075.jpg" /> blows up at<img src="8-4500105\1ad8d5e8-ecea-4ea1-a58f-5e985249e618.jpg" />, as mentioned before, the curvature components of the Rimennian tensor appear to behave (under the assumption M &gt; 0) perfectly normally at<img src="8-4500105\bbf77f16-70de-4be7-b9e5-e7ced6e342ba.jpg" />. Further, the determinant of the metric coefficients continues to be negative and finite<img src="8-4500105\b07213ab-782a-4785-9f12-886203442c19.jpg" />. Such realizations gave rise to the idea that the Schwarzschild coordinate system suffers from a “coordinate singularity” at the event horizon and must be replaced by some other well behaved coordinate system. It is only in 1960 that Kruskal and Szekeres [9,10] discovered a one-piece coordinate system which can describe both the interior and exterior regions of a BH. They achieved this by means of the following coordinate transformation for the exterior region (Sector 1):</p><disp-formula id="scirp.26225-formula143286"><label>(2)</label><graphic position="anchor" xlink:href="8-4500105\4c932c98-69ac-4cbb-9095-efd3e7b4c96d.jpg"  xlink:type="simple"/></disp-formula><p>where</p><disp-formula id="scirp.26225-formula143287"><label>(3)</label><graphic position="anchor" xlink:href="8-4500105\8643e84a-ed1a-4396-b2e4-3cd24c4185aa.jpg"  xlink:type="simple"/></disp-formula><p>It would be profitable to note that</p><disp-formula id="scirp.26225-formula143288"><label>(4)</label><graphic position="anchor" xlink:href="8-4500105\ecf07bce-b82b-4813-bcfb-c115dcc4538c.jpg"  xlink:type="simple"/></disp-formula><p>And for the region interior to the horizon (Sector 2), we have</p><disp-formula id="scirp.26225-formula143289"><label>(5)</label><graphic position="anchor" xlink:href="8-4500105\a82b9bbe-a188-4138-9929-2d97cc9e1783.jpg"  xlink:type="simple"/></disp-formula><p>where</p><disp-formula id="scirp.26225-formula143290"><label>(6)</label><graphic position="anchor" xlink:href="8-4500105\7a130c6f-a138-4339-ab8e-544df6bb0c19.jpg"  xlink:type="simple"/></disp-formula><p>and</p><disp-formula id="scirp.26225-formula143291"><label>(7)</label><graphic position="anchor" xlink:href="8-4500105\2751fcd2-bd78-40bc-8a02-b455e7b729dd.jpg"  xlink:type="simple"/></disp-formula><p>Note these are the only coordinates which involve <img src="8-4500105\d4871749-51b0-45fd-879f-e6000e886d89.jpg" /> in the denominator. Given our adopted signature of spacetime<img src="8-4500105\55af267d-09eb-4bf6-95f9-9ac55ed654bf.jpg" />, in terms of <img src="8-4500105\3a4721c6-d3ea-4c2a-935c-64cd5623e433.jpg" /> and<img src="8-4500105\4f167c91-ee9d-4ceb-bb6b-733c6b37cf49.jpg" />, the metric for the entire spacetime is</p><disp-formula id="scirp.26225-formula143292"><label>(8)</label><graphic position="anchor" xlink:href="8-4500105\2f31917b-7c53-4e28-914c-fe4a8e1fbabc.jpg"  xlink:type="simple"/></disp-formula><p>The metric coefficients are apparently regular everywhere except at the intrinsic singularity<img src="8-4500105\332cba07-33f9-4dff-9a2e-ec1563146a4f.jpg" />. Note that, the angular part of the metric remains unchanged by such transformations and <img src="8-4500105\c33f4b59-1851-450d-9e2a-6d77bf76e045.jpg" /> continues to signal its intrinsic spacelike nature. In either region we have</p><disp-formula id="scirp.26225-formula143293"><label>(9)</label><graphic position="anchor" xlink:href="8-4500105\a7905ece-5a24-47b4-91a4-49ab36e3a27e.jpg"  xlink:type="simple"/></disp-formula><p>so that</p><disp-formula id="scirp.26225-formula143294"><label>(10)</label><graphic position="anchor" xlink:href="8-4500105\07564e77-7082-4ef8-9d4a-81d43aa6f230.jpg"  xlink:type="simple"/></disp-formula><disp-formula id="scirp.26225-formula143295"><label>(11)</label><graphic position="anchor" xlink:href="8-4500105\3c5b7db9-5e5f-4e9b-ab02-e607c522a3ea.jpg"  xlink:type="simple"/></disp-formula><p>and</p><disp-formula id="scirp.26225-formula143296"><label>(12)</label><graphic position="anchor" xlink:href="8-4500105\9690f805-3d90-48f7-b404-0c22877025eb.jpg"  xlink:type="simple"/></disp-formula><p>So, very strangely, the <img src="8-4500105\9ddaed69-ccb6-4b08-b58d-de304ac74b67.jpg" /> point corresponds to not one but two conditions!</p><disp-formula id="scirp.26225-formula143297"><label>(13)</label><graphic position="anchor" xlink:href="8-4500105\158e8650-1b7b-4b58-8ec7-36eb1fd01bf7.jpg"  xlink:type="simple"/></disp-formula><p>Here, one point needs to be hardly overemphasized; astronomical observations and experiments actually conform to the idea that atleast far from massive bodies or even BH Candidates (BHC), the spacetime is well described by the <img src="8-4500105\f6d35978-d59e-4775-8324-50910a569846.jpg" /> coordinate system. In fact, although in the (normal) physical spacetime, in a spherically symmetric spatial geometry (as defined by the implications of <img src="8-4500105\72224f81-539d-4bb7-a79a-3a45d32ac073.jpg" /> as an “invariant circumference radius”), the physical singularity corresponds to a mathematical point, in the Kruskal world view, this central singularity corresponds to a pair of hyperbolas in the <img src="8-4500105\a1b9aa12-5fc0-4045-9224-6c228b7a9c73.jpg" /> plane. While the “+ve” sign of equation corresponds to the central BH singularity, the “−ve” sign corresponds to the singularity inside a so-called White Hole which may spew out mass-energy spontaneously in “our universe” [6,7]. The white hole singularity belongs to “other universe” whose presence is suggested by the fact that the Kruskal metric remains unaffected by the following additional transformations:</p><disp-formula id="scirp.26225-formula143298"><label>(14)</label><graphic position="anchor" xlink:href="8-4500105\4997587c-687b-4090-a760-9c446c66887d.jpg"  xlink:type="simple"/></disp-formula><p>defining Sector (III) and</p><disp-formula id="scirp.26225-formula143299"><label>(15)</label><graphic position="anchor" xlink:href="8-4500105\b06f99e2-1d3e-4825-8d90-58d4f600f79f.jpg"  xlink:type="simple"/></disp-formula><p>defining Sector 4. Thus not only does the region interior to the EH correspond to two different universes (Sectors 2 and 4), but the structure of the physical spacetime outside the EH, too, effectively corresponds to two universes (Sectors 1 and 3). Hence, if there would be <img src="8-4500105\a245c32a-6dcc-45fe-a4b0-bbe075ada79d.jpg" /> separate BHs, as per the Kruskal prescription, there would be <img src="8-4500105\863c2ca0-70fa-4c56-b5ca-33cb8734d32a.jpg" /> disconnected physically weird universes describing different wormholes and other universes. Hence if a massive star would indeed collapse to form BH in the universe we live, other universes would be instantly born!</p></sec><sec id="s3"><title>3. Kruskal Derivative</title><p>The aim of this paper is to explicitly verify whether the (radial) geodesics of material particles are indeed timelike at the EH, which they must be if this idea of a finite mass Schwarzschild BH is physically correct. First note that if the test particle is released from rest at<img src="8-4500105\8d7e369f-395a-411b-8626-2fca70527b6b.jpg" />, then the conserved mass-energy per unit rest mass <img src="8-4500105\031ccaf4-1ad8-4a95-a85f-79bbc115e9f2.jpg" /> is given by</p><disp-formula id="scirp.26225-formula143300"><label>(16)</label><graphic position="anchor" xlink:href="8-4500105\0086c637-5e68-4b7c-98bd-fbf8fc9f8491.jpg"  xlink:type="simple"/></disp-formula><p>Note, from various considerations, in Pap. I [<xref ref-type="bibr" rid="scirp.26225-ref8">8</xref>], we have already found that, at the EH, one should have</p><disp-formula id="scirp.26225-formula143301"><label>(17)</label><graphic position="anchor" xlink:href="8-4500105\0c471ae7-1b1b-4f1b-9fb1-9ae33d456b94.jpg"  xlink:type="simple"/></disp-formula><p>Armed with this value of<img src="8-4500105\4ab9bfd5-6aaf-4080-b463-45fb90f5be72.jpg" />, we are in a position now to complete our task by rewriting the radial part of the Kruskal metric<img src="8-4500105\11a4f0b7-8060-423c-8601-b2a69bf4daba.jpg" /> as</p><disp-formula id="scirp.26225-formula143302"><label>(18)</label><graphic position="anchor" xlink:href="8-4500105\f1b31737-bd4e-48a6-bc79-e43b2fdeeab6.jpg"  xlink:type="simple"/></disp-formula><p>or,</p><disp-formula id="scirp.26225-formula143303"><label>(19)</label><graphic position="anchor" xlink:href="8-4500105\6fcf6ea9-ecce-4699-b65a-bdb11fb8a558.jpg"  xlink:type="simple"/></disp-formula><p>This implies that although the metric coefficients can be made to appear regular, the radial geodesic of a material particle would become null at the event horizon of a finite mass BH in contravention of the basic premises of GTR! Therefore GTR must not allow occurrence of any EH at all [2,3]!</p><p>In fact this result could have been easily anticipated by studying the limiting behaviour of the SBH metric in Schwarzschild coordinates too (even if one would assume that Schwarzschild coordinates break down at exact<img src="8-4500105\cf3d8b33-ccf0-4024-8be6-946df7b016c9.jpg" />). Using the vacuum Schwarzschild metric, for a radial geodesic in, one finds that find that [2,3]</p><disp-formula id="scirp.26225-formula143304"><label>(20)</label><graphic position="anchor" xlink:href="8-4500105\4199068d-0aed-4151-b992-352b3215ac95.jpg"  xlink:type="simple"/></disp-formula><p>i.e.,</p><disp-formula id="scirp.26225-formula143305"><label>(21)</label><graphic position="anchor" xlink:href="8-4500105\b9fa1d26-a899-4d7b-a99a-7533abbda55d.jpg"  xlink:type="simple"/></disp-formula><p>But since</p><disp-formula id="scirp.26225-formula143306"><label>(22)</label><graphic position="anchor" xlink:href="8-4500105\06ee3121-5c0f-45f7-8bc8-b1fc7dc43fa7.jpg"  xlink:type="simple"/></disp-formula><p>for a radial geodesic, one eventually finds that</p><disp-formula id="scirp.26225-formula143307"><label>(23)</label><graphic position="anchor" xlink:href="8-4500105\0a67fa67-e9a5-4657-84cc-b9f9ed9ed480.jpg"  xlink:type="simple"/></disp-formula><p>Note that for a photon, <img src="8-4500105\76534da3-fad5-4983-87f3-d37aca315cce.jpg" />, and the foregoing equation correctly shows that<img src="8-4500105\da2c450e-eaa1-4308-9411-57ba46331ac4.jpg" />. Also, for a material particle with<img src="8-4500105\aee4340c-a9cd-43d9-89ff-af2faf7d35d0.jpg" />, it shows that indeed <img src="8-4500105\8b81864f-ac6b-4d9a-b924-18c1effa1544.jpg" /> as long as<img src="8-4500105\d8230207-8c6f-4166-868b-8928b724f9b1.jpg" />. However even for a material particle worldline,</p><disp-formula id="scirp.26225-formula143308"><label>(24)</label><graphic position="anchor" xlink:href="8-4500105\9ce02999-0744-4a3c-bd86-4b594408b4f7.jpg"  xlink:type="simple"/></disp-formula><p>irrespective of the value of<img src="8-4500105\362cc46e-7d8b-4293-a674-733e79b77556.jpg" />.</p><p>Therefore, the radial geodesic of a material particle in the Schwarzschild metric would become unphysically null <img src="8-4500105\f1e143a7-9949-4e94-8cb5-c32c04daef70.jpg" /> in case the particle would arrive at the EH. It may be noted here that though<img src="8-4500105\0631f31b-dbd0-4af1-80e7-3d45a1ca28f8.jpg" />, by definition <img src="8-4500105\d4c82d5b-b333-4fa4-a0a0-c6e29203fb7a.jpg" /> is an infinetisimal quantity. And since <img src="8-4500105\b178b405-137c-4e30-a37f-844521814b05.jpg" /> is an invariant, its limiting value must not depend on the coordinates used. Accordingly, it is natural that Kruskal coordinates too should lead to <img src="8-4500105\2f0c8d34-f1af-40cf-a869-026c007461cc.jpg" /> as<img src="8-4500105\c49679e4-10af-428b-9f77-1a4b1c2cb2f1.jpg" />.</p><p>Note, <img src="8-4500105\2ff4ac17-a23d-48cc-a6b1-996e11fd6f2d.jpg" />can be obtained from several approaches, and in Pap. I, we found that <img src="8-4500105\ed495ae4-bc15-4ca5-9b82-a9c38ce3d252.jpg" /> form in certain cases. Accordingly, let us evaluate <img src="8-4500105\5275c910-b107-469c-861c-a98dc1e14f31.jpg" /> for such apparent <img src="8-4500105\9d7e166c-4c4f-46f8-ab98-8f7e56f436e3.jpg" /> cases more carefully.</p><sec id="s3_1"><title>3.1. <img src="8-4500105\93f99b84-5935-48a3-b087-eefd489b2cd4.jpg" />Form of the Kruskal Derivative</title><p>First let us define a quantity</p><disp-formula id="scirp.26225-formula143309"><label>(25)</label><graphic position="anchor" xlink:href="8-4500105\57d7c1d0-7ad5-43d2-9f42-f82e67a79da8.jpg"  xlink:type="simple"/></disp-formula><p>In Pap. I [2,3,8], by the brute direct approach, we found that</p><disp-formula id="scirp.26225-formula143310"><label>(26)</label><graphic position="anchor" xlink:href="8-4500105\edbe70f3-08f6-461a-8d30-70e1a451c87d.jpg"  xlink:type="simple"/></disp-formula><p>To evaluate this derivative very carefully, close to the EH, let us first write</p><disp-formula id="scirp.26225-formula143311"><label>. (27)</label><graphic position="anchor" xlink:href="8-4500105\ba554417-4824-4a4b-82c4-13d6cb04d446.jpg"  xlink:type="simple"/></disp-formula><p>Clearly, the value of <img src="8-4500105\1c598c2f-6874-4bc3-ae53-7261af61b83b.jpg" /> depends on whether <img src="8-4500105\b1229107-9e93-4701-ac99-c0651d443237.jpg" /> or<img src="8-4500105\dc1da434-f6a4-4389-9005-ffb1ce80c4a1.jpg" />, and let us first consider the former case:</p></sec><sec id="s3_2"><title>3.2. <img src="8-4500105\bbd474e3-4304-4c1c-94d9-e37114d9ded3.jpg" />Case</title><p>When<img src="8-4500105\b2d32d32-cb00-45c0-af02-5e711f95f1ae.jpg" />, we have</p><disp-formula id="scirp.26225-formula143312"><label>. (28)</label><graphic position="anchor" xlink:href="8-4500105\d3f79e99-52aa-4b57-adf7-43f158243f5d.jpg"  xlink:type="simple"/></disp-formula><p>And thus close to the EH,</p><disp-formula id="scirp.26225-formula143313"><label>. (29)</label><graphic position="anchor" xlink:href="8-4500105\afaf253c-221b-486d-a318-14c73821fd91.jpg"  xlink:type="simple"/></disp-formula><p>For the positive sign of<img src="8-4500105\dfab4a4b-d65a-446f-b1b9-50175cf905fb.jpg" />, as<img src="8-4500105\b62edd6a-ba7d-4405-8323-c5ff422e775b.jpg" />, one finds</p><disp-formula id="scirp.26225-formula143314"><label>. (30)</label><graphic position="anchor" xlink:href="8-4500105\326c511f-cdca-425d-bbb5-f53734178acc.jpg"  xlink:type="simple"/></disp-formula><p>While for the negative sign of<img src="8-4500105\e2fe7f61-f558-4d76-ba8b-d67bd203a21d.jpg" />, one has</p><disp-formula id="scirp.26225-formula143315"><label>. (31)</label><graphic position="anchor" xlink:href="8-4500105\9228bb97-1c29-4c7e-ac53-aed24f7d0df8.jpg"  xlink:type="simple"/></disp-formula><p>Thus, for<img src="8-4500105\efae0b26-701f-4dc7-a57f-7985ff2bfc33.jpg" />, <img src="8-4500105\43cce1c9-21a5-46fd-b96a-e23e754db9d4.jpg" />as obtained in Pap I.</p></sec><sec id="s3_3"><title>3.3. <img src="8-4500105\4417dbba-3bcc-4aee-a609-795970743e1d.jpg" />Case</title><p>For<img src="8-4500105\0186b200-3ec6-4f0d-8adc-d9e51894a83d.jpg" />, close to the EH, one has</p><disp-formula id="scirp.26225-formula143316"><label>(32)</label><graphic position="anchor" xlink:href="8-4500105\ff7c1dd7-ec38-465d-a66c-c6782c1732d9.jpg"  xlink:type="simple"/></disp-formula><p>and</p><disp-formula id="scirp.26225-formula143317"><label>(33)</label><graphic position="anchor" xlink:href="8-4500105\9699407c-8a84-4928-ac96-626d560cd5a7.jpg"  xlink:type="simple"/></disp-formula><p>where</p><disp-formula id="scirp.26225-formula143318"><label>. (34)</label><graphic position="anchor" xlink:href="8-4500105\59505402-7132-4019-9ffa-ad41f1d9daa0.jpg"  xlink:type="simple"/></disp-formula><p>Simultaneously from (9), one finds that</p><disp-formula id="scirp.26225-formula143319"><label>(35)</label><graphic position="anchor" xlink:href="8-4500105\3fcb024c-5c94-4ac5-af8d-6a0abeaf7680.jpg"  xlink:type="simple"/></disp-formula><p>so that</p><disp-formula id="scirp.26225-formula143320"><label>(36)</label><graphic position="anchor" xlink:href="8-4500105\734bdc15-d75d-4ee9-a659-2ac57bbaaa6c.jpg"  xlink:type="simple"/></disp-formula><disp-formula id="scirp.26225-formula143321"><label>. (37)</label><graphic position="anchor" xlink:href="8-4500105\3ed92519-4593-4de1-88cb-fec5241e460a.jpg"  xlink:type="simple"/></disp-formula><p>Thus, one has <img src="8-4500105\b33e708a-0d5f-4a77-84de-b045827e7c6c.jpg" /> sets of expressions for the value of <img src="8-4500105\b06117b6-45df-45c9-b6f3-906ffe84fef1.jpg" /> at<img src="8-4500105\83e250bd-1ad2-48b4-bbef-19a0584e4618.jpg" />. And as we retain terms only upto first order of<img src="8-4500105\cb3c0142-170a-41c0-86ac-53e1e8393ea9.jpg" />, and further let <img src="8-4500105\b75cce62-1383-4fb0-b70e-1452c5d48cde.jpg" /> whenever appropriate, from (26) we obtain:</p><p>1) <img src="8-4500105\6389fa85-bb22-4d37-be2a-ca7108bf37bb.jpg" />and <img src="8-4500105\60e86be7-b2dd-44a8-9517-65cb4acfc621.jpg" /></p><disp-formula id="scirp.26225-formula143322"><label>(38)</label><graphic position="anchor" xlink:href="8-4500105\6ce3fc3f-166c-4400-b2fa-a09c19731641.jpg"  xlink:type="simple"/></disp-formula><p>By using the forms of <img src="8-4500105\88638620-6997-40f2-b48d-999adcbc2ebb.jpg" /> and<img src="8-4500105\7e2f0d7c-f97e-486d-b42e-a27aa46d89ae.jpg" />, the above expression can be seen as</p><disp-formula id="scirp.26225-formula143323"><label>(39)</label><graphic position="anchor" xlink:href="8-4500105\125acc8b-87f7-4394-b151-cb96e85bc69f.jpg"  xlink:type="simple"/></disp-formula><p>If<img src="8-4500105\4118b4eb-60fc-45b9-8d3c-955928139be6.jpg" />, then one will have<img src="8-4500105\2260c2e8-9091-499f-b0e0-fc537b2ab013.jpg" />. But, if one will have<img src="8-4500105\0678f40e-22c3-4ef8-8853-cb087209fa28.jpg" />, then, once again</p><disp-formula id="scirp.26225-formula143324"><label>(40)</label><graphic position="anchor" xlink:href="8-4500105\31fbba3c-27aa-41cf-b676-ea8de81c1694.jpg"  xlink:type="simple"/></disp-formula><p>2) <img src="8-4500105\64127386-78ad-4028-ae5b-e063dba3e44a.jpg" />and <img src="8-4500105\cf2bede7-fab8-4bff-86bc-9a97fce14721.jpg" /></p><disp-formula id="scirp.26225-formula143325"><label>(41)</label><graphic position="anchor" xlink:href="8-4500105\c1a595e5-afe5-4718-81f6-9807c4ae3c9c.jpg"  xlink:type="simple"/></disp-formula><p>Again, by using the forms of <img src="8-4500105\2011ceb0-5ccc-4aa3-8391-443534ff9b66.jpg" /> and<img src="8-4500105\b571b5c1-2ee6-40d5-af43-7affc8ba1eca.jpg" />, the above expression can be seen as</p><disp-formula id="scirp.26225-formula143326"><label>(42)</label><graphic position="anchor" xlink:href="8-4500105\cacadcce-54c6-4dc4-b732-65af1cc8c20a.jpg"  xlink:type="simple"/></disp-formula><p>If<img src="8-4500105\6d504918-5158-416d-9d7c-15f059ee98c6.jpg" />, then one will have<img src="8-4500105\faf1e6be-0b76-4a24-a9af-42ba40a10422.jpg" />. But, if one will have<img src="8-4500105\e735b68f-c2fc-4970-a6d2-6b2a9411c97f.jpg" />, then, once again</p><disp-formula id="scirp.26225-formula143327"><label>(43)</label><graphic position="anchor" xlink:href="8-4500105\d718edbc-ca29-4839-af09-40708cd48126.jpg"  xlink:type="simple"/></disp-formula><p>3) <img src="8-4500105\1ee0d8d0-2d8f-48e5-8c20-8c93f547a006.jpg" />and <img src="8-4500105\3d1679c0-3611-4245-a0d4-2bed04f2366d.jpg" /></p><disp-formula id="scirp.26225-formula143328"><label>(44)</label><graphic position="anchor" xlink:href="8-4500105\95e97372-ea8f-47d2-8841-f6b704e8be0e.jpg"  xlink:type="simple"/></disp-formula><p>4) <img src="8-4500105\02102af9-dce0-48d0-8ce8-0d034a250f8a.jpg" />and <img src="8-4500105\3edf3dcb-b594-4572-8d2f-bdbf67a56ab2.jpg" /></p><disp-formula id="scirp.26225-formula143329"><label>(45)</label><graphic position="anchor" xlink:href="8-4500105\9c7fc82c-7fe1-44de-b136-8faf3f7e18ea.jpg"  xlink:type="simple"/></disp-formula><p>Thus when we consider <img src="8-4500105\aacf0e21-46f4-413e-8f89-1a4ac6ff11cb.jpg" /> and<img src="8-4500105\7c07a77a-f774-4716-adb6-8426d2b81650.jpg" />, we obtain self-contradictory and strange results. Apparently the cases 1) &amp; 2) suggest that <img src="8-4500105\2c2f7b01-f466-4317-b597-6dd3c1db02f3.jpg" /> provided <img src="8-4500105\2f428d64-d2a4-4ac0-bad1-78904604fbe9.jpg" /> so that <img src="8-4500105\50e03ebc-a520-4c0f-875a-1ba263987ab3.jpg" /> even as<img src="8-4500105\ed913d06-dfda-4b4a-a458-ebe056ed8ece.jpg" />. Note that <img src="8-4500105\43ad20be-7f7d-4a80-9df3-b3443bd2e2ad.jpg" /> is an invariant, and thus such a result would be in contradiction with Equation (23) which shows that <img src="8-4500105\4351aefc-4b6a-4137-a4df-9936248919ab.jpg" /> as <img src="8-4500105\a0128a2c-8961-4164-90cf-3bcb47675579.jpg" /> irrespective of the value of<img src="8-4500105\024abfd5-d18e-4ad9-bf9a-841f3a17ede7.jpg" />. Indeed the hope that <img src="8-4500105\46ce0bd6-8516-4438-b320-b738d0bbf0e3.jpg" /> is belied in cases 3) &amp; 4), which yield <img src="8-4500105\26934643-14dd-4f06-abed-bf13470f9d26.jpg" /> as before. And therefore, for a graceful exit from such a self-contradiction, one must accept that <img src="8-4500105\3943dac6-804d-4553-9e51-083fc60df856.jpg" /> as <img src="8-4500105\799af10e-5932-43c8-9006-d4e83f8bf987.jpg" /> whose physical interpretation is <img src="8-4500105\b81647c0-7099-42cd-bc75-ea050f9500b4.jpg" /> for a point particle.</p></sec></sec><sec id="s4"><title>4. Interpretation</title><p>In view of such inconsistencies for <img src="8-4500105\275040e1-27d0-47d0-ad68-954411a20fc9.jpg" /> and <img src="8-4500105\afc49bc8-0f92-414d-9409-383beecc3b6e.jpg" /> case, it appears that the presumption, <img src="8-4500105\55785bbc-69e5-45f7-93c0-00c6f15a6756.jpg" />, on which the BH paradigm is based is incorrect. And even if we would momentarily accept this incorrect presumption, we would have to confront with the result that as the EH would be reached, <img src="8-4500105\52a25d89-2d06-45c6-a0f9-fd5d19eb46fe.jpg" />, i.e., geodesic would become lightlike. But this possibility is inconsistent with the BH paradigm where the EH is a regular region of spacetime and where the geodesic of a material particle must be timelike, i.e.,<img src="8-4500105\1574b414-741b-436e-b5c3-233df9c05027.jpg" />. Since <img src="8-4500105\c1605833-eecc-4bdc-863d-bef7ea2c8a36.jpg" /> is an invariant, we can not blame the coordinate system to be faulty for such an inconsistency. So we must reject the presumption that a neutral point particle appearing in the vacuum SBH solution has a finite gravitational mass.</p><p>Therefore, clearly, for self-consistency, as far as a point mass is concerned (but not necessarily for an extended object like the Sun), one should have only<img src="8-4500105\6c4cfb94-13de-4887-b4f1-8b8d0c5b3889.jpg" />. And since, <img src="8-4500105\f44fe212-77d8-4467-b176-20995b60dece.jpg" />can very much be finite, it appears from Equation (16) that, for a neutral point mass or a SBH, one must have<img src="8-4500105\988bc326-3942-484f-9692-f1a2329d589a.jpg" />.</p><p>For the SBH, there is only one spacetime singulatity which is at<img src="8-4500105\e0df9fc9-450b-4a3d-80f4-b17bc20445f4.jpg" />. Therefore, the EH and the true physical singularity must be the same:</p><disp-formula id="scirp.26225-formula143330"><label>(46)</label><graphic position="anchor" xlink:href="8-4500105\d3553ce5-2232-4500-8ee5-3e64e9dbae6b.jpg"  xlink:type="simple"/></disp-formula><p>And thus for the true SBH, one must have</p><disp-formula id="scirp.26225-formula143331"><label>(47)</label><graphic position="anchor" xlink:href="8-4500105\c8f8d96e-1ae2-4d59-998a-f89941e76dd1.jpg"  xlink:type="simple"/></disp-formula><p>Irrespective of the above interpretation, the very fact the radial geodesic would turn lightlike if it would hit the EH means that the EH is no regular region of the spactime; on the other hand, the EH must be a spacetime singularity where the “once timelike always timelike” mathematical rule would break down. This is similar to the special relativistic situation where a test particle having a timelike geodesic would turn lightlike if there would be an accelerator of infinite capacity which would push<img src="8-4500105\401e88e4-59c6-4d8c-8bfd-a4a56379bda0.jpg" />. In contrast, if finite mass SBHs would really have been allowed, one should have had <img src="8-4500105\a9116192-f094-43e4-b95c-fb3a59da59c9.jpg" /> irrespective of the value of <img src="8-4500105\06955b93-a737-4814-a5d3-50705fc709f8.jpg" /> and irrespective of the coordinates employed. Indeed, the occurrence of <img src="8-4500105\c581cdea-3421-475d-a2d1-ad21863b7bdd.jpg" /> as<img src="8-4500105\78dc2712-c92d-4e11-82ee-309dbccea6aa.jpg" />, proves in a coordinate independent manner that for any physically meaningful definition of 3-speed<img src="8-4500105\bbdb4bb1-4213-41c7-a5f7-8431ad085479.jpg" />, one must have <img src="8-4500105\d585f896-b80a-4e2d-82fd-48f4b08d7278.jpg" /> as <img src="8-4500105\bf4a3886-676a-49dc-b738-e7528ce01389.jpg" /> [2,3] irrespective of the coordinates used. The latter fact was even acknowledged by Crawford &amp; Tereno [<xref ref-type="bibr" rid="scirp.26225-ref11">11</xref>] after initially denying it:</p><p>“Since this is an unacceptable result and we know that the Schwarzschild coordinate system is not suitable for describing the manifold at <img src="8-4500105\7cf958ab-565f-4161-b615-60e1019204b8.jpg" /> it is rather tempting to blame the coordinate system for this malfunction. But we should ask first, could it be possible to find a coordinate system that does not have this defect? The answer is obviously no, since the result is independent of the choice of coordinates, &#183;&#183;&#183; Indeed, even if we use a coordinate system that has no difficulties at<img src="8-4500105\ced9a6a5-6833-4396-ad70-d1fd6ea0e5f3.jpg" />, like the advanced Eddington-Finkelstein coordinates, we would still end up with the same result <img src="8-4500105\9da48287-1cec-4c4a-a91f-ad0ccbd646f2.jpg" /> as<img src="8-4500105\1b00a150-6d42-43c1-aa55-6552f2a55bdd.jpg" />.”</p><p>However, yet, to bypass this fact, they demanded that the speed of one infalling particle/observer having <img src="8-4500105\e9d1aa5e-4581-423b-a77b-69fccf25804c.jpg" /> must be measured by another infalling particle/ observer having <img src="8-4500105\d10738e7-752e-48a7-b499-a29c58de3140.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref11">11</xref>]. This essentially means that to see whether a speeding car is violating some speed limit or not, the traffic inspector too must travel alongside the car rather than sit at the checkpost! By adopting this strange prescription for checking speed, they found that, the relative 3-speed of the infalling particles, with respect to each other, is given by[<xref ref-type="bibr" rid="scirp.26225-ref11">11</xref>]</p><disp-formula id="scirp.26225-formula143332"><label>(48)</label><graphic position="anchor" xlink:href="8-4500105\f7a93ec4-7a13-4529-8f6b-ac2828fff750.jpg"  xlink:type="simple"/></disp-formula><p>where<img src="8-4500105\54cce798-a5f6-403a-adc5-2fc341489a94.jpg" />, (note, in different context, we also use<img src="8-4500105\7c458898-23bf-4b4a-a451-f5c58d65bb5e.jpg" />). And at the EH, this yields</p><disp-formula id="scirp.26225-formula143333"><label>(49)</label><graphic position="anchor" xlink:href="8-4500105\f2b73940-109b-451b-94cd-15e185b2cd7a.jpg"  xlink:type="simple"/></disp-formula><p>sothat, apparently, <img src="8-4500105\52263d5c-b915-48ba-991c-60967e9ccb51.jpg" />if<img src="8-4500105\0d015415-f2f6-4058-8ad4-7a61b59db74b.jpg" />. In particular, if<img src="8-4500105\bdffce0b-8550-45d1-bc39-9d53a9a808a9.jpg" />, one will have<img src="8-4500105\9382673e-4af4-46f9-a8d0-90f7477d0b51.jpg" />; i.e., the observer/particle would conclude that he/she is at rest at the EH. This is a profound self-contradiction because Crawford &amp; Tereno starts with the premises that the concept of staticity vanishes at the EH and below it. In fact, in GR, the local speed must indeed be measured locally; i.e., the observer and the test particle must be at the same spatial location. In other words, the two infalling particles must have the same comoving proper time at a given spatial position. For a free fall, this can be ensured everwhere, only when the two particles/observers have the same initial conditions, i.e.,<img src="8-4500105\37cc718c-e7a8-4e3a-84a4-ba4a7c9592e0.jpg" />. Thus as per Crawford &amp; Tereno, the observer must fall along with the test particle side by side! And when this is so, one must have <img src="8-4500105\49e09b24-6dde-49d7-8934-c7902140cac8.jpg" /> everywhere! This is like the situation, when the speeding car driver dictates that the traffic inspector must measure his speed by sitting in the same car, in the adjacent seat [<xref ref-type="bibr" rid="scirp.26225-ref3">3</xref>]! With reference to Equation (48), even if one would assume<img src="8-4500105\b33c390a-a291-4bf1-9d11-0fb7fe52e724.jpg" />, and<img src="8-4500105\39275b17-1165-47e4-9ac1-f6a6d47a3ba4.jpg" />, at<img src="8-4500105\7d5ea423-65ff-4367-b9e8-247b06f011b7.jpg" />, one will have <img src="8-4500105\53200763-4360-4a3a-9c9f-0c9bbffec73f.jpg" /> because</p><disp-formula id="scirp.26225-formula143334"><label>(50)</label><graphic position="anchor" xlink:href="8-4500105\d2eeac0b-3d46-4e97-9014-fb264215ea06.jpg"  xlink:type="simple"/></disp-formula><p>Therefore, following Crawford &amp; Terneno prescription, even at the central singularity</p><disp-formula id="scirp.26225-formula143335"><label>(51)</label><graphic position="anchor" xlink:href="8-4500105\d0566330-3989-4835-ae08-787c5e4dd792.jpg"  xlink:type="simple"/></disp-formula><p>And this above absurdity can be removed by considering that at the EH, <img src="8-4500105\3b65b623-cc3f-4d91-bd7b-021240d3a9c7.jpg" />so that<img src="8-4500105\1c833d82-e3aa-460a-8a67-d7342f8ed618.jpg" />.</p><p>Unfortunately, the same physically nonsense ansatz for speed measurement has recently been adopted by another author too [<xref ref-type="bibr" rid="scirp.26225-ref12">12</xref>]. He [<xref ref-type="bibr" rid="scirp.26225-ref12">12</xref>] has attempted to include angular momentum in the ansatz of Crawford &amp; Tereno. For certain conditions, he indeed finds that: “So even if it had not go to r = 0 the particle is seen to move with the velocity of light.” But he proceeds by ignoring this redsignal. He again finds, “At large value of angular momentum the velocity approaches that of 1” (p. 12). But strangely he opines the <img src="8-4500105\3a0699f5-523d-4baf-b28e-6c837e37e0ac.jpg" /> situation to be justified!</p><p>Even when one considers accretion of a perfect fluid onto a SBH [<xref ref-type="bibr" rid="scirp.26225-ref13">13</xref>]“calculations show that the total velocity <img src="8-4500105\16d82adb-e128-4c31-be6f-0618f3583fa1.jpg" /> tends toward <img src="8-4500105\433da0ef-3aa1-48aa-9cb9-b02105928ca5.jpg" /> as<img src="8-4500105\b0075bd7-5472-4dbe-a263-e2c4bc517052.jpg" />, which is in accordance with expectations” (see p. 9) [<xref ref-type="bibr" rid="scirp.26225-ref13">13</xref>].</p><p>Unfortunately, these authors fail to realize here that ocurrence of <img src="8-4500105\859cb866-dabb-4211-bf18-2fa7a9da0b21.jpg" /> is not allowed in GR because this implies <img src="8-4500105\e262cc9c-83ff-4410-8d49-e7537c4aa067.jpg" /> for the accreting fluid! On the other hand, such results actually imply that astrophysical and all other so-called BHs must be something else [14,15].</p><p>Let us also ponder, how a test particle can be prevented from hitting the EH ever in order that its timelike geodesic remains so. The test particle would never arrive at the EH, if the associated proper time would be infinite: i.e.,</p><disp-formula id="scirp.26225-formula143336"><label>(52)</label><graphic position="anchor" xlink:href="8-4500105\730f26f7-6dfe-41d2-beeb-5308597f1725.jpg"  xlink:type="simple"/></disp-formula><p>Clearly this demands that the gravitational mass of the SBH<img src="8-4500105\8fbb5607-a226-4f36-be35-85b5fdf2b02b.jpg" />.</p><p>Note, when one incorrectly presumes a BH with<img src="8-4500105\7b5405e6-d9fe-4428-bf97-73574d129744.jpg" />, one arrives at a fundamental self-contraction: As per the distant inertial observer (who actually does the experiment), the test particle can approach the EH only asymptotically, and can neither ever reach it or penetrate it. On the other hand, as per the free falling observer, he not only arrives at the EH but can reach the central singularity too. Such a dichotomy is against the principle of general covariance by which ultimate physical results must not depend on choice of coordinates, and must be same for all observers. And this fundamental inconsistency gets resolved only when one realizes that for a BH, <img src="8-4500105\18231c9c-349c-4cd0-8a0e-31807fa9ad66.jpg" />, so that both<img src="8-4500105\aa50b839-360b-4c00-8c2a-9ca0f6321ea4.jpg" />.</p><p>Also note, unlike the case of Newtonian gravity, in GTR, <img src="8-4500105\f8d7b361-034b-4d8e-a5e9-5766bbc50a8d.jpg" />state need not correspond to a configuration with zero baryonic mass. The <img src="8-4500105\c2c5ef4d-c29c-4feb-a320-e7d4b001e840.jpg" /> state is simply one in which the negative gravitational energy exactly offsets the positive energy associated with <img src="8-4500105\09fd85db-62cc-4d7d-9549-89e4d62d4d37.jpg" /> and internal energy, and may indeed represent a physical singularity with infinite energy density and tidal acceleration.</p><p>On the other hand, since in Newtonian gravity, the negative self-gravitational energy does not offset the bare mass, a point particle can have arbitrary large positive mass; and hence, the concept of a finite mass BH fits better in a Newtonian context.</p></sec><sec id="s5"><title>5. Discussions</title><p>Even the authors who, in a desperate bid to save the black hole paradigm, invent almost ridiculous definition of “free fall speed” by which an infalling particle has <img src="8-4500105\ae22f2c0-a48b-44c8-b139-ed21d0496a92.jpg" /> even at the central singularity, were forced to conclude that [<xref ref-type="bibr" rid="scirp.26225-ref16">16</xref>]</p><p>“The solutions that do away with the interior singularity and the event horizon, although interesting in themselves, sweep the inherent conceptual difficulties of black holes under the rug. In concluding, we note that the interior structure of realistic black holes have not been satisfactorily determined, and are still open to considerable debate.” (Emphasis by the author).</p><p>If the staple topic of discussion of lakhs of textbooks and may be millions of articles and news reports have not been “satisfactorily determined” in almost 100 years, it is certain that, the paradigm itself is unphysical and faulty.</p><p>On the other hand, with the direct result that <img src="8-4500105\4986ef86-976e-4306-9c87-63ae03c75cd3.jpg" /> for the SBH, the entire conundrum of “Schwarzschild singularity”, swapping of spatial and temporal characters by <img src="8-4500105\90ebaf35-92f3-49b3-8e28-75999bd6745f.jpg" /> and <img src="8-4500105\e895b3e2-d211-436e-9098-b1a849216681.jpg" /> inside the event horizon (when the angular part of all metrics suggest that <img src="8-4500105\588156f3-5e80-418a-b72c-4164970e71f6.jpg" /> has a spacelike character even within the horizon), “White Holes” and “Other Universes” get resolved. Here we recall the wise comments of Rosen [<xref ref-type="bibr" rid="scirp.26225-ref17">17</xref>].</p><p>“So that in this region <img src="8-4500105\efdabad0-cfa6-4d83-a7fd-247ccab5c9b5.jpg" /> is timelike and <img src="8-4500105\ed844536-0f1e-420d-a462-f941f2edab67.jpg" /> is spacelike. However, this is an impossible situation, for we have seen that <img src="8-4500105\2032c266-19ec-441a-bf98-eff12bd54913.jpg" /> defined in terms of the circumference of a circle so that <img src="8-4500105\d0b1af1b-210a-4c17-98e1-dc05e4c2e11f.jpg" /> is spacelike, and we are therefore faced with a contradiction. We must conclude that the portion of space corresponding to <img src="8-4500105\8a6cd2bc-ae09-4b29-93b4-6b2c885adaa7.jpg" /> is nonphysical. This is a situation which a coordinate transformation even one which removes a singularity can not change. What it means is that the surface <img src="8-4500105\334a77c5-9ac6-4978-8953-a91bec2ecafd.jpg" /> represents the boundary of physical space and should be regarded as an impenetrable barrier for particles and light rays.”</p><p>This idea of Rosen is also in accordance with the idea of Einstein that the Schwarzschild type singularity is unphysical and can not occur for realistic cases [<xref ref-type="bibr" rid="scirp.26225-ref18">18</xref>].</p><p>And this paper indeed shows that in order that the radial worldlines of free falling material particles do not become null at a mere coordinate singularity, Nature (GTR) refuses to have any spacetime within the EH. And this unphysical happening is of course avoided when we realize that <img src="8-4500105\24df3f1d-6046-4e6a-b5f4-335c88049db1.jpg" /> and there is no additional spacetime between the EH and the central singularity.</p><sec id="s5_1"><title>5.1. Nature of Black Hole Candidates</title><p>Some readers would however ignore this cogent result that <img src="8-4500105\766ca184-c399-4e84-bfbb-47cabe721110.jpg" /> for a true BH, by arguing that there are massive compact astrophysical objects and which must be BHs. Such an argument would be based on the fact that cold self-gravitating objects cannot be more massive that few solar masses<img src="8-4500105\71df24b1-f03a-49f5-8205-67f9a97425ac.jpg" />in view of Chandrasekhar and Tolman OppenheimerVolkoff (TOV) limits.</p><p>Note the concepts of Chandrasekhar or TOV are based on degenerate fermions at temperature<img src="8-4500105\d82d3262-edfe-447d-886a-d5f70e3922cd.jpg" />. Therefore, they are irrelevant for objects which are extremely hot and not supported by mere cold degenerate pressure. For instance there known hot stars with masses as large<img src="8-4500105\e1e5fb1d-c10b-4cc3-9353-72a6a8a2f313.jpg" />, and in principle, there could be radiation pressure supported stars (RPSSs) with masses as large as <img src="8-4500105\0df93dae-4dd8-43bd-827f-9d4a2a9d299c.jpg" /> [7,19]. Such quasi-Newtonian RPSSs however have a lower mass limit of <img src="8-4500105\17b84024-678b-4c37-97d6-83462574e437.jpg" /> and they are considered as strictly static solutions. Accordingly, they must obey the Buchdahl limit <img src="8-4500105\bce55d9e-1dd5-4d68-9d94-607d3491096b.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref20">20</xref>]. Indeed such quasi-Newtonian supermassive stars in principle may have a surface gravitational red-shift <img src="8-4500105\b9c2e955-2b5f-4760-b59d-7ad22760babf.jpg" /> [7,16]. In contrast, in principle there could be quasi-static extremely relativistic radiation pressure supported stars with <img src="8-4500105\179abbbb-aabd-40bb-a364-c9b4a71fd64f.jpg" /> (RRPSSs) and no lower mass limit as well [21-23]. It may be recalled that it was Hoyle &amp; Folwler who first suggested that the central compact objects of quasars could be hot quasi-Newtonian RPSSs [24,25]. They however conceived these RPSSs as strictly static ones whose source of internal energy and pressure is due to central nuclear burning; i.e., they ignored the fact quasistatic RPSSs can generate internal energy simply by virtue of quasi-static gravitational contraction [21-23].</p><p>In fact there are several other alternative BH models and all of which may have <img src="8-4500105\5f98a900-d801-4cb1-b0b0-2fed6cb22ef2.jpg" /> and arbitrary high mass [<xref ref-type="bibr" rid="scirp.26225-ref26">26</xref>]. It is also often argued that, the observed BH candidates must be true BHs, because, radio and X-ray observations might have probed them down to few Schwarzschild radii. Clearly, in view of the existence of several BH candidates with <img src="8-4500105\2523029a-9f4e-4faa-8266-18d17cb507f9.jpg" /> and<img src="8-4500105\dabdc73d-db1c-478b-93a3-2d5bd5593ed0.jpg" />, such astronomical observations have not at all confirmed that the so-called BH candidates are true BHs. Further it is hoped that the Event Horizon telescope would actually image the EH of the BH candidates. In reality, because of strong gravitational lensing near an ultracompact object, the distantly observed image would always be larger than the photon sphere having <img src="8-4500105\9ba17963-18de-4a4f-b190-e41a16a658bf.jpg" /> and<img src="8-4500105\56e75738-efc9-4c27-a6ac-a875fb36d62f.jpg" />. Thus no telescope would ever be able to detect the fictitious EH. And of course, by definition, it is not possible to detect the EH “from which nothing, not even light can escape”.</p><p>Hence there is no observational proof which can mysteriously upstage the exact result <img src="8-4500105\713e635e-eadf-4bcf-9e55-64d202a6ba9f.jpg" /> obtained here.</p></sec><sec id="s5_2"><title>5.2. Gravitational Collapse</title><p>Some readers may argue that there is an exact GR solution by Oppenheimer &amp; Snyder (OS) which shows that sufficiently massive objects must undergo gravitational collapse to form BHs, and hence the integration constant appearing in SBH or Kruskal solution cannot be zero. The OS solution assumes the collapsing object to be homogeneous when no self-gravitating object can be strictly homogeneous [<xref ref-type="bibr" rid="scirp.26225-ref27">27</xref>]. Further, it assumes the collapsing matter to be “dust with no pressure at all;<img src="8-4500105\3e560667-d050-49df-a4a5-8adea10a7b02.jpg" />. If the homogeneity assumption would be dropped, even the fictitious pressure-less dust solutions often lead to EH less Naked Singularity rather than a BH [<xref ref-type="bibr" rid="scirp.26225-ref4">4</xref>]. And it has recently been shown that the OS solution, though may appear to be mathematically correct, has only a symbolical value because it actually corresponds to zero matter density <img src="8-4500105\48475266-cc0a-48c3-bcc7-7f530632a67b.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref28">28</xref>]. This is expected because exact zero pressure, can be achieved only mathematically when <img src="8-4500105\2e750cd1-6217-429d-a7d6-21e32cc311c7.jpg" /> too. Thus, in reality, there is no exact GR solution which indicates formation of finite mass BHs. Given the physical fact that the strict <img src="8-4500105\0eaf91ec-5f1d-4aac-9e2b-bbad3b40a43c.jpg" /> condition implies<img src="8-4500105\54a5821d-e8dc-46bc-8515-166f9dac73f8.jpg" />, many of the examples of naked singularity formation in GR collapse are equally fictitious. Even if one would ingore the proof that for a dust<img src="8-4500105\b877522b-869b-4588-bdbe-0467a9d61ba0.jpg" />, a critical analysis of Oppenheimer-Snyder collapse made in the Schwarzschild frame has revealed that, in reality, OS collapse does not result in any event horizon, any trapped surface or any finite mass black hole [<xref ref-type="bibr" rid="scirp.26225-ref29">29</xref>]. And this profound result is based on the simple mathematical fact that the argument of a logarithmic function must be positive definite [<xref ref-type="bibr" rid="scirp.26225-ref29">29</xref>].</p><p>On the other hand, realistic gravitational collapse always involves pressure gradient, heat and radiation transport [1-3,21-23]. When such complexities are invoked, there is no general exact solution of the problem, and to make progress, one has to make various simplified and favourable assumptions. Since, one never knows beforehand which of such simplifications are valid, one cannot make any general claim as to whether continued GR collapse gives rise to BHs or Naked Singularities, or some other non-singular objects. As to the various claims about the occurrences of naked-singularities, we may recall [<xref ref-type="bibr" rid="scirp.26225-ref30">30</xref>].</p><p>“Although some theoretical counterexamples have been constructed, the general consensus is that these are all too artificial too occur naturally.”</p><p>In fact, same is true for all claims of BH formation in continued gravitational collapse too. In order to claim that, a spacetime singularity has been formed, it is not enough to find whether some light rays can escape outside before being trapped (on which claims of naked singularities are based). On the other hand, one must faithfully compute the comoving proper time all the way <img src="8-4500105\b59e0046-4905-4451-ac2f-2c384ea69f8a.jpg" /> to confirm that<img src="8-4500105\22289ccf-0286-400a-9701-543a662db594.jpg" />. Such a computation is possible only for the fictitious dust case, and for no physically realistic case. Thus all claims of formation of “naked singularities” are non-sound.</p><p>In fact by virtue of the general proof that trapped surfaces are actually not formed [31,32], it is most likely that continued collapse results in non-singular objects having external radius <img src="8-4500105\4fe74f15-07a8-43d1-984c-fc941946b4c3.jpg" /> [2,3]. The proof for non-occurrence of trapped surfaces was also offered by Kriele [<xref ref-type="bibr" rid="scirp.26225-ref33">33</xref>]. Ironically, such a non-occurrence of trapped surfaces and non-formation of finite mass BHs in continued collapse might be mistaken as evidence for formation of naked singularities. In the absence of a trapped surface or EH, such objects must keep on radiating and contracting even if at infinitesimally slow rate. In fact there are several GTR special solutions for physical gravitational collapse which suggest that effect of radiation pressure and dissipation may cause formation of hot radiating ultra-compact objects rather that any BH or naked singularity [34,35]. Radiation pressure apart, generation of tangential pressure too can arrest the continued collapse.</p><p>As the contracting objects would keep on losing mass energy, it is likely that, they would approach the <img src="8-4500105\8f7803e2-10ec-45e8-ad96-b81de17fb867.jpg" /> true BH state. However, this state must not be allowed to be formed in finite commoving proper time, i.e., ever. This must be so because even for a<img src="8-4500105\afdc9e24-4c4c-44b3-b1ec-842deff78f07.jpg" />, BH, the timelike geodesic of an infalling material particle would tend to turn lightlike if the particle would ever arrive at the EH. Therefore, the most logical scenario seems to be one where continued gravitational collapse would indeed continue indefinitely (Eternally Collapsing Object: ECO) [22,23,31,32,36]. Since astrophysical plasma is always associated with imbedded magnetic field, ECO are expected to be ultra-magnetized with a pulsar like magnetosphere around. Thus it is quite likely the GR collapse results in the formation of Magnetospheric Eternally Collapsing Objects (MECOs).</p></sec></sec><sec id="s6"><title>6. Conclusions</title><p>Even if one would prima-facie accept the BH paradigm which is based on the presumption that the integration constant appearing in the vacuum Schwarzschild solution <img src="8-4500105\76e72297-deff-4889-98a5-a5a93462f59d.jpg" /> is finite for a massenpunkt or a “point mass” too, one lands up in many puzzles and unphysical happenings. For instance, one must wonder, if the EH is indeed a perfectly non-singular regular region, why would it require an infinite upward boost for an object to stay put there [<xref ref-type="bibr" rid="scirp.26225-ref6">6</xref>]. And such an infinite boost is required irrespective of whether one is using much maligned Schwarzschild coordinates or any other supposedly well behaved coordinates. Clearly, the requirement of an infinite boost and the property that nothing, not even light can escape the clutches of gravity at the EH are very much physical aspects, and they signify that the EH is a physical singularity. Yet, one overlooks such physical questions to defend the BH paradigm by inventing various other coordinate systems to somehow hide the physical problems associated with the EH. And here we considered the Kruskal coordinates which are believed to be the ultimate tool to establish the BH paradigm. As soon as the Kruskal extensions were proposed, they raised even more physical questions like the apparent existence of other universes, wormholes etc., and some authors pointed out some of the unphysical aspects. For instance Anderson &amp; Gautreau [<xref ref-type="bibr" rid="scirp.26225-ref37">37</xref>] pointed out that the Kruskal scheme may involve causal violations even at<img src="8-4500105\a3bd7d31-d8a8-45c8-a6a5-c51f8da76681.jpg" />. Belinfante hinted at some of the weird predictions of the Kruskal scheme [<xref ref-type="bibr" rid="scirp.26225-ref38">38</xref>]. Later Gautreau concluded that [<xref ref-type="bibr" rid="scirp.26225-ref39">39</xref>].</p><p>“I give arguments showing that the reference system is not maximally extended, as is commonly reported in the literature. On both Novikov and Kruskal Szekeres spacetime diagrams, the left-hand side, corresponding to negative values of the spatial coordinate, should not be included when describing a physical spacetime. In turn, this means we have to rethink widely-accepted concepts such as black and white holes that arise from the usual picture of a maximally-extended Kruskal Szekeres spacetime”.</p><p>Antoci &amp; Liebscher [<xref ref-type="bibr" rid="scirp.26225-ref40">40</xref>] pointed out that EH is actually a physical singularity and the weird picture of Kruskal coordinates is not realizable.</p><p>However what these authors have not pointed out is a much simpler and profoundly fundamental incongruity associated with the Kruskal coordinates. Note the original problem of the static central gravitational field, i.e., the Schwarzschild solution, is solved by using the natural assumption that the spacetime is asymptotically flat. Here, for<img src="8-4500105\63fedbc5-ed55-4fc8-9eb7-016cb746f728.jpg" />, one must recover the Minkowski metric:</p><disp-formula id="scirp.26225-formula143337"><label>(53)</label><graphic position="anchor" xlink:href="8-4500105\8b5e545f-6a70-46fb-bc45-ef8f4e00df88.jpg"  xlink:type="simple"/></disp-formula><p>Very oddly, the Kruskal solution is NOT asymptotically flat.</p><p>As<img src="8-4500105\b8cfdf82-6ab4-49fc-8d32-413d97e21c30.jpg" />, the Kruskal metric becomes</p><disp-formula id="scirp.26225-formula143338"><label>(54)</label><graphic position="anchor" xlink:href="8-4500105\2de00ef2-fc06-4197-9a82-030ca0d83662.jpg"  xlink:type="simple"/></disp-formula><p>One may recoincile this with the previous equation only by presuming <img src="8-4500105\dc669d8f-f218-4cae-bdbd-1971d458dec2.jpg" /> at <img src="8-4500105\219d7e26-5403-434a-8e5b-d93061dd895e.jpg" /> when in the original problem, <img src="8-4500105\126566fa-6e4f-4e6a-aeba-a80f85e8c7a9.jpg" />at<img src="8-4500105\a146403d-f4e4-432f-a92f-d734b4801471.jpg" />. Thus Kruskal metric brazenly contradicts the foundations of the very problem it purported to solve. Actually, even in the limit, <img src="8-4500105\9109a16c-0b33-458c-bca6-aec15fe40781.jpg" />, the skewed “Kruskal manifold is topologically different from the Minkowski manifold” [<xref ref-type="bibr" rid="scirp.26225-ref40">40</xref>].</p><p>As far as Differential Geometry and mathematics are concerned, one can indeed conceive of complex manifolds which cannot be covered by a single coordinate chart and may possess any number of strange properties. Such exercises could delight many mathematicians and keep them absorbed; but that does not mean that, the observable physical spacetime must be such complex, convoluted, strange and often self-contradictory. For instance, one may easily conceive of not ony 5-D but 26-D or any dimensional spacetime, and arrive at billions of exact solutions. Many such solutions could also suggest existence of even stranger kind of BHs, wormholes and what not. But such mathematical extravaganza need not have any relationship with the observable physical world.</p><p>Even for the innocuous 4-D GTR, most of the “exact solutions” could be physically misleading, and actually vacuous. For instance, it was recently found that, the innocuous simple problem of a strictly uniform density self-gravitating sphere is vacuous, because it actually corresponds to <img src="8-4500105\7e78dd5e-ea54-4b13-b4a3-bf2a62d5c400.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref27">27</xref>]. Similarly, the exact solutions associated with the collapse of this homogeneous sphere too vacuous as they correspond to <img src="8-4500105\5d6c9ac6-5d5e-451c-8978-38ef126eada3.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref41">41</xref>]. And the collapse of a homogeneous pressure-less collapse (the OS collapse) too is vacuous because it corresponds to <img src="8-4500105\acb5f59a-97ee-44db-8d8c-39986a175af9.jpg" /> [28,29]. As recently found, one of the most important metrics in GR, namely the de-Sitter metric, which is the basis for supposed “cosmic inflation” and “dark energy” is illusory because, for self-consistency, one must have cosmological constant <img src="8-4500105\0e4f78eb-d576-48e0-a298-51ee28f1b274.jpg" /> [<xref ref-type="bibr" rid="scirp.26225-ref42">42</xref>].</p><p>In GR, one is in principle free to use arbitrary coordinates; but the use of complex, convoluted coordinates invented by one mathematician after another need not lead to new physical realities. Similarly, while coordinate transformations could sometimes be mathematically convenient, they themselves must not lead to new physical realities. Accordingly, the very idea that the Kruskal coordinates are the most ideal coordinates and reveal various universes, white holes, worm holes etc. and which are not revealed by other coordinates are against the spirit of the principle of general covariance. Only if a specific coordinate can be associated with geometrically or physically relevant quantities, it may be considered as a better coordinate to represent the inherent physical reality despite the principle of covariance. And the Kruskal coordinates <img src="8-4500105\ed038c12-5ada-4b53-8592-52189ec47dc0.jpg" /> by no way could be related to any physical observables. As we just found, they even blatantly contradict the basic fact that the concerned spacetime is asymptotically flat. Note, the Kruskal coordinates <img src="8-4500105\f2282972-fd5a-4e66-bdb1-158d476813bf.jpg" /> are constructed by using Schwarzschild/Hilbert coordinates<img src="8-4500105\2f8d7a6a-5d6b-49e9-9704-c0c56a06fd55.jpg" />. And if <img src="8-4500105\6e7d1705-b164-415b-8f18-80f1629be63d.jpg" /> are “bad” coordinates, how can the coordinates made out of them could be “good” coordinates?</p><p>We feel that the reason that the Kruskal coordinates give an infinite physical distortion is that they involve division by zero (see <img src="8-4500105\daea9be6-c0e6-4efa-8f64-17a8726614d0.jpg" /> and <img src="8-4500105\75665a2d-b4ce-4f75-bf8a-f7566ea09547.jpg" /> terms in Equations 2-3), because the mass of the point particle <img src="8-4500105\71696883-8535-4fad-afc6-8b178700be75.jpg" /> has been found to be zero. In contrast, the Schwarzschild radial coordinate <img src="8-4500105\a7fdef10-f48d-46f2-ad0a-59e7ed0818f2.jpg" /> has direct geometrical significance because, by definition, <img src="8-4500105\1563f19d-6195-4c65-a94e-d9712a0fb721.jpg" />, represents the invariant area of 2-surfaces around the centre of symmetry. In view of such an invariant character, <img src="8-4500105\da1ab917-f274-4cc1-8d0d-7354cd109254.jpg" />defines the luminosity distance too. Similarly, the Schwarzschild coordinate <img src="8-4500105\dae8d6e3-4002-4182-9134-5e3f888bb30c.jpg" /> has a solid physical implication as the proper time measured by a distant inertial observer. Thus in reality, the Schwarzschild coordinates are the most appropriate coordinates for studying a problem having a spherical symmetry. (Note, this so-called “Schwarzschild coordinate” <img src="8-4500105\98e95d6a-3a02-4bc0-be9f-246e6e0781c9.jpg" />and the metric are originally due to Hilbert, and not due to Schwarzschild). On the other hand, these physically significant coordinates appear not to cover the interior region of a fictitious BH whose idea crops up when one one incorrectly presumes that even a neutral point particle has finite gravitational mass. In contrast, in Newtonian gravitation, a point particle may be assumed to have arbitrary mass. And as far as GR is concerned, there may not be any strictly point particle at all.</p><p>The great expectation that the Kruskal extension represents a new and complete physical picture was rightly dismissed by Dirac in 1962 [<xref ref-type="bibr" rid="scirp.26225-ref43">43</xref>]:</p><p>“The mathematicians can go beyond this Schwarzschild radius, and get inside, but I would maintain that this inside region is not physical space, because to send a signal inside and get it out again would take an infinite time, so I feel that the space inside the Schwarzschild radius must belong to a different universe and should not be taken into account in any physical theory.”</p><p>From a different consideration, Kiselev, Logunov, &amp; Mestvirishvili too have shown that finite mass BHs are in contradiction with GTR [<xref ref-type="bibr" rid="scirp.26225-ref44">44</xref>]. Further long back Narlikar &amp; Padmabhan too noted many conceptual difficulties associated with the concept of “Event Horizon” and “Black Hole” [<xref ref-type="bibr" rid="scirp.26225-ref45">45</xref>]:</p><p>“Nevertheless there are several conceptual difficulties associated with this simple and elegant solution that are usually ignored because of its manifest usefulness.”</p><p>“For the detection of any object by whatever means, it must come within the observer’s past lightcone. This does not ever happen for a BH. So none of the laws describing the behavior of BHs (as opposed to the Quasi BHs) are in principle detectable or testable by the class of observers who stay outside their event horizons. Since most observers (including those on the Earth) are of this type, to them the BH’s are not relevant as physical objects” [<xref ref-type="bibr" rid="scirp.26225-ref45">45</xref>].</p><p>“We therefore find that considering observers inside the event horizon makes the problems of interpretation even more difficult, and we wonder whether nature allows gravitational collapse to continue inside the event horizon at all” [<xref ref-type="bibr" rid="scirp.26225-ref45">45</xref>].</p><p>Indeed the SBH solution is exact and beautiful; and therefore, the resolution to all such paradoxes can be made only when we realize that while the integration constant <img src="8-4500105\7fd2a81d-ab89-4846-82f3-59512accd89e.jpg" /> is indeed finite for an extended object, it shrinks to zero as the radius of the object shrinks to zero. In 1969, Bel noted that [<xref ref-type="bibr" rid="scirp.26225-ref46">46</xref>]:</p><p>“Actually, several extensions have been proposed in the literature, the most commonly quoted being those of Finkelstein and Kruskal. Both extensions lead to spacetime models which are not globally static and are consequently inadequate for representing the exterior solution of a source in static equilibrium.”</p><p>Then by considering intrinsic differential geometry associated with the problem, Bel concluded that [<xref ref-type="bibr" rid="scirp.26225-ref46">46</xref>].</p><p>“Schwarzschild singularity becomes instead a real point singularity on which are localized the sources of the exterior static solution” (Emphasis by the author).</p><p>This means that the point particle is synomous with the Event Horizon, as has been repeatedly stressed by this author while being unaware about Bel’s ignored conclusion.</p><p>Of course, one obtains BH like solutions in many other gravity theories and all quantum gravity theories too. But everywhere it must be the same story; the BH solutions represent asymptotic static limits of dynamical solutions characterized by <img src="8-4500105\2ff4634a-4af8-430b-95e4-b5ad254fe0e2.jpg" /> and never realizable in physical world. Thus mathematical studies of BHs and questions like how such vacuum solutions can possess huge entropies and micro-states are only idle mathematical exercises without any physical content. In view of the asymptotic <img src="8-4500105\a7052996-7f6d-491f-a164-2d9968c683ca.jpg" /> BH solution, we realize that “point particles/ singularities” are never allowed by GTR even though the concept of such a “point particle” is required for mathematical tractability. Given this, the concept of elementary “strings” or “branes” seem to be important ones; such concepts, by definition, eliminate the point singularities. But ironically the super-string theories too take the appearances of static BH solutions seriously when the very concept of extended “strings” and “branes” are anti-thesis of singularities!</p><p>Irrespective of the present study, there have been already direct proof that the integration constant appearing in the vaccuum Schwarzschild/Hilbert solution is zero [31,32,47,48]. Thus the massive compact BH candidates cannot be true BHs. Indeed there are significant amount of observational evidences, that the so-called BH candidates could be ultracompact, ultramagnetized, ultrahot balls of plasma [49-55]. Recall that the Sun too is a magnetized plasma and the violent eruptions like Coronal Mass Ejection (CME) could be traced to such magnetized plasma ativities. Similarly, much of the violent activities associated with the so-called BH candidates may be triggered by relativistic version of CMEs from the ultra-magnetized plasma of MECOs. Even if uncharged BHs would be assumed to be spinning, they are electromagnetically inert because no current can flow out of the central singularity, and no energy can be extracted [<xref ref-type="bibr" rid="scirp.26225-ref56">56</xref>]. On the other hand, spin down energy can be extracted from a spinning MECO as it would act like an ultra-relativistic pulsar [<xref ref-type="bibr" rid="scirp.26225-ref57">57</xref>].</p><p>Very recently, the Event Horizon Telescope has imaged a plasma jet coming out from a region within<img src="8-4500105\1a0dce80-f173-4575-a059-33b2b8ec5025.jpg" />. Since it is bent by the gravitation of the central compact object [<xref ref-type="bibr" rid="scirp.26225-ref58">58</xref>], it is highly likely that this jet is emanating from the compact object rather than from its accretion disk. If so, this could be almost a direct confirmation that the central object here is a MECO rather than a true BH with an EH. Recall, solar prominences and coronal mass ejection from the Sun are plumes or bursts of plasma emerging from the magnetized plasma of the sun. Similarly, the plumes of plasma imaged from near the compact object of M87 is most likely the signature of a Magnetospheric ECO which is spewing out magnetized plasma irrespective of any accretion disk activity.</p><p>We note here that the faith in the non-existent “Event Horizon” and the pretention to be unaware about the relevant developments, have driven theoretical physics in a vicious blind alley. When Quantum Gravity is supposed to remove the BH singularity, it has actually been invoked to enhance the notion of BHs. On the one hand, physical reality of BHs are insisted by claiming that “the EH is a regular space-time in VACUUM, a mere coordinate singularity, with no physical effects”. On the other hand, the same EH is modeled sometimes as “membranes”, sometimes as “fluids”, sometimes as “Fuzzball”, sometimes as hard “brick walls”, and now as “Firewalls” [<xref ref-type="bibr" rid="scirp.26225-ref59">59</xref>] with most dazzling physical effects! And they never admit that they are running into such brazen contradictions because the very assumption of finite mass BHs is incorrect. As one can recall, these all started with endowing the avowed vacuum EH, a non-physical “coordinate singularity”, with magical thermodynamic properties and then inventing “Hawking Radiation” from there. In reality, even by mundane classical GTR, as we found here, there is no finite mass BH [28,31,32,47,48] there is no EH, no EH thermodynamics (except area A = 0, and entropy S = 0), no “Hawking Radiation”, and no voodoo “Firewall” [2,3,23,28,31,47]. Further instead of fictitious quantum gravitational “Hawking Radiation” or preHawking radiation, any gravitational collapse is accompanied by well understood radiation in the form of photons and neutrinos for which no unfounded QG is required [1-3,21-23,36].</p><p>Similarly, hypothetical magical effects associated with the claimed non-physical “coordinate singularity” have inspired formulation of mystic “Emergent Gravity” [<xref ref-type="bibr" rid="scirp.26225-ref60">60</xref>] and “Entropic Gravity” [<xref ref-type="bibr" rid="scirp.26225-ref61">61</xref>] speculations. One may of course imagine that the vacuum is meshed by elementary Planck scale cells; but that does not mean that such cells are moving randomly like the molecules of a real gas. Recall, as per the basic premises of Quantum Mechanics, if the Planck Length would tend to zero, one must recover classical results by which vacuum has zero entropy. But as per the “Emergent Gravity” formulation, the entropy of classical vacuum would be infinite (as Planck const &#224; 0)! This shows complete unphysical and incorrect nature of such popular speculations. Atleast, Padmanabhan could have avoided such speculations by recalling his own honest conclusion:</p><p>“The discussion of physical behavior of black holes, classical or quantum, is only of academic interest” [<xref ref-type="bibr" rid="scirp.26225-ref45">45</xref>].</p></sec><sec id="s7"><title>7. Endnote</title><p>A preprint by the same title (arXiv:astro-ph/9904162v1) has been there on Cornell Univ. Preprint ArXiv for the past 13 years. The fact that, this preprint has not received any criticism for 13 long years implies that its basic content is correct. The present version is a massive revision of the same preprint, it encompasses all the related developments that have taken place over 13 years; and practically, this is a highly updated new paper. It is nice to see that two anonymous referees of IJAA verified/ corrected all the relevant calculations.</p></sec><sec id="s8"><title>8. Acknowledgements</title><p>The author thanks the anonymous referees for pointing out some errors/typos in the initial two versions of this manuscript. He also thanks IJAA for waiving the processing fee in view of the importance of this paper.</p></sec><sec id="s9"><title>REFERENCES</title></sec></body><back><ref-list><title>References</title><ref id="scirp.26225-ref1"><label>1</label><mixed-citation publication-type="other" xlink:type="simple">A. Mitra, “Why Gravitational Contraction Must Be Accompanied by Emission of Radiation in Both Newtonian and Einstein Gravity,” Physical Review D, Vol. 74, No. 2, 2006, Article ID: 024010.  
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