Criticism of the clarification on Schwarzschild solution by Christian Corda (3)

This writer claims that the original Schwarzschild solution is equivalent to the standard Schwarzschild solution and that the difference between them is merely a matter of different coordinate systems. This, of course, is contrary to reality because Schwarz's original solution is based on a singularity point at the center of mass, while the standard solution results in a singularity point at a certain distance from the center. The writer has tried to evade this clear truth with arguments that contain a lot of effort and affectation that do not get rid of the contradiction but move it to another field. He claims, for example, that the point of origin of the coordinate system in the original solution is not a point, and that the relationship between the radius and the circumference of the circle in this system It is not the simple, well-known relationship...etc. It is known that changing the reference system cannot turn the point into a sphere, cannot change the relationship between the radius and the circumference of the circle, and cannot change the geometry of the space. Changing coordinate systems does not do any of this, but it only changes the analysis of quantities into their dimensions. For example, the distance between two points remains in its shape and length regardless of the reference system with which we look at it, but it only differs from one system to another in the way it is analyzed into its three dimensions.

Logical Analysis of Schwarzschild's Solutions (1)

In this article, I want to compare the different treatments of the issue of the exact solution of the Einstein equation in terms of its logical structure and to clarify the relationships between the results reached by these treatments. This comparison should be accompanied by a criticism of some of the contents of these treatments, especially the incorrect application of the principle of general covariance . It is advisable to start first with the general framework of the solution, which is agreed upon by everyone who dealt with this issue, starting with Schwarzschild, then Droste and Hilbert, and everyone who came after them. Understanding this general framework is very important in this comparison because it distinguishes for us between the real areas of disagreement that we are looking for and the detailed mathematical issues that do not matter for physicists very much like the different methods for solving differential equations and the mathematical tricks used to simplify certain processes. A Schwarzschild Solution (in its general sense, which includes the modified solutions that came later)begins by writing down the functions according to which the parts of the metric are distributed around a center in which there is a mass. These functions are originally 16 functions, but after we apply to them the rules of general symmetry and then the conditions specific to our problem, which are spherical symmetry and stationary conditions we reach a small number of unknown functions, so we try to determine them by applying Einstein's equation and our physical information about the gravitational field. Then, if that is not enough to determine the unknown functions, we ask for help from other sources, which are the ones in which the treatments differed, and let us call them "auxiliary sources". Based on this, we can divide the different treatments of this issue into three sections: The first: Schwarzschild's treatment, the auxiliary source of which was an intelligent guess (but there is no guarantee that it will be correct) of the location of the singularity. Schwarzschild used this source and arrived at a specific solution ( Not that one under his name found in textbooks but the one found in his original paper). Second: The treatment of Droste, Hilbert and others, which is still present in important references by great physicists such as Dirac and Weinberg. The auxiliary source in these treatments is (unfortunately and surprisingly) the incorrect application of the principle of general covariance to the laws of physics. In this treatment, one of the unknown functions is eliminated in the following way: Suppose that one of the unknown functions s f(r), so if you replace it with r , then what you did will not change the problem or its form because you only changed the reference system, and this is correct, but what is not correct is to think that we thus have gotten rid of one of the unknowns. In fact, in this case, we transferred our ignorance of the function to the meaning of variables that we are talking about . After we continue solving the equations and reach the final solution, we find that it is written in quantitative terms similar in symbol to the radius that we all know, but in reality (and this is what the references that rely on this method do not say) it is only unkown function in radius. The third section: The treatment that does not need any auxiliary source, as it takes advantage of all the conditions of spherical symmetry more efficiently than the Schwarzschild and Hilbert treatment, so that the number of unknown functions is only two. This treatment can be found in many modern scientific papers, but I have not been guided and have not yet searched for its first source. The conclusion that we will detail in future articles, God willing, is that Schwarzschild’s solution is completely logically sound, but its result is wrong because his guess about the location of the singularity was not correct, and that Droste and Hilbert’s solution has a problem with incorrect use of the covariance , but nevertheless the result they arrived at is correct. The correct solution in terms of logic, physics of relativity, and the final result is the last section.

تعليق على فحص حل شفارزشيلد الأصلي لمعادلة آينشتين الذي كتبه محمد المنصور حسني ( أو حسّاني)

هذه الصورة توضح أهم جزء من مقاله المنشور في موقع ڤيكسرا وغيره.

(معذرةً لعدم استعمال علامات الترقيم بسبب مشكلة أرجو أن تُحل لاحقا) وقد ذكر في مقاله كثيرا من الكلام الجيد في بعض العموميات مثل الفرق بين الفيزياء والرياضيات إلا أن اعتراضه الذي اعترض به على عمل شفارزشيلد وأسس عليه المقال لم يكن اعتراضا صحيحا بل سوء فهم منه لخطوة شفارز وتطبيق لقاعدة التحليل البعدي في غير محلها لاحظ شفارزشيلد أن محددة المتري في حالة الاحداثيات القطبية العادية غير ثابتة ولا تساوي الواحد الصحيح ولاحظ أنه لو استعمل بدل هذه الاحداثيات احداثيات أخرى تكون محددتها مساوية لواحد في كل مكان فإن ذلك سيبسط المسألة كثيرا لأنه سيلغي بعض الحدود المعقدة في تكوين ممتد ريتشي الداخل في المعادلة وهنا اهتدى شوارز لحيلة ذكية وهي أن يغير الاحداثيات القطبية العادية الى احداثيات قطبية أيضا حتى لا يتغير شكل المتري لكنها مختلفة عنها وهو بهذا يعتمد على المعنى العام للاحداثيات القطبية وهي كل الاحداثيات المبنية على نقطة أصل تنطلق منها محاور في كل الاتجاهات ليست بالضرورة أن تكون أنصاف الأقطار العادية بل أي دالة فيها و كذلك أن يفصل بين كل محور وما يجاوره من المحاور قيمة تعتمد على الزاوبة بينهما وليس بالضرورة الزاوية نفسها لكن أي دالة فيها وهذا ما التبس على كاتب المقال وربما بسبب أن شفارزشيلد رمز لنظام المحاور الجديد برموز تشبه رموز الاحداثيات الكارتيزية فظن أن الاحداثيات هي احداثيات كارتيزية وهذا ظاهر عندما طبق عليها نظرية التحليل البعدي فإنه أسند لها كلها أبعاد الطول على أنه حتى لو وجد اختلاف في الأبعاد بين الاحداثيات الأصلية والاحداثيات المحول اليها فهذه ليست مشكلة أبدا لأن معادلة التحويل من احداثي الي احداثي اخر ليست علاقة فيزيائية بين كميتين بل اختلاف في الطريقة التي نريد أن نحسب بها المسافة فمثلا لو فرضنا اننا نبحث في مسألة بالاحداثيات الكارتيزية المتعامدة المعتادة ثم أراد شخص ما أن يستعمل احداثيات أخرى هي نفس هذه الاحداثيات مع استبدال المحور س بمحور آخر هو س تكعيب فلن توجد أي مشكلة في هذه الاحداثيات الجدبدة ويمكن أن نطبق كل نظريات الفيزياء على الكميات المحسوبة من خلالها ولا يصح أن نعترض عليه بأن أبعاد س تختلف عن أبعاد س تكعيب. أراد شوارز أن يستعمل احداثيات تبسط الحسابات ثم يعود بالنتيجة التي وصل اليها ويترجمها الى الاحداثيات القطبية العادية التي بدأ بها المسألة بالنسبة لهذه النقطة فإن خطوة شوارزشبلد سليمة وفيها ذكاء وابداع يحسب له وهذا لا ينفي أنه في خطواته اللاحقة وقع فيما أوجب التصحيح والمراجعة وذلك عندما افترض مسبقا أن موضع الشذوذ في الحل الذي يطلبه سيكون في نقطة الأصل ومعلوم الآن أنه اذا حسب بالطريقة الصحيحة فإنه في نقطة غير نقطة الاصل تعرف بنصف قطر شوارز يعتمد على الكتلة فبنيت عليه نظرية الثقوب السوداء ولكن هذا ليس موضوع هذا التعليق.

Comments on Scrutiny of Schwarzchild's Original Solution by Mohamed E. Hassani (2)

In his article, he mentioned a lot of good things about some generalities, such as the difference between physics and mathematics. However, his objection to Schwarzschild’s work, upon which the article was based, was not a valid objection, but rather a misunderstanding on his part of Schwarz’s step and a misplaced application of the rule of dimensional analysis. Schwarzschild noted that the determainant of the metric In the case of ordinary polar coordinates, is not constant and do not equal one. Then he noted that if instead of these coordinates other coordinates were used whose determinant is equal to one everywhere, this would greatly simplify the problem because it would eliminate some of the complex terms in the formation of the Ricci tensor included in the equation, and here Schwarz was guided by a clever trick. It is to change the normal polar coordinates to another polar coordinates as well so that the metric general form does not change, but it is different from them in detais . In this way, he depends on the general meaning of polar coordinates, which are all coordinates based on a point of origin from which axes radiate in all directions. They do not necessarily have to be the ordinary radii, but rather any function of them. Likewise, each axis is separated from its neighboring axes by a value that depends on the angle between them and not necessarily the angle itself, but any function in it, and this is what confused the author of the article, and perhaps because Schwarzschild used symbols for the new axis system similar to the symbols of Cartesian coordinates, he thought that the new coordinates are Cartesian coordinates, and this is apparent when he applied the theory of dimensional analysis to them. He assigned to them all dimensions of length, although even if there were found a difference in dimensions between the original coordinates and the new coordinates then this is not a problem at all because the equation of converting from one coordinate to another is not a physical relationship between two quantities, but rather a difference in the way we want to calculate the distance. For example, if we assume that we are investigating a problem with the usual Cartesian orthogonal coordinates, then a person wants use another coordinates that are the same as these coordinates but replaces the x-axis with another axis equal to cubic x, he can use them without any problem and apply the same laws of physics on the quantities written according to thiese coordinates and we should not then say that this disagree with the theory of dimensional analysis. Schwarz wanted to use coordinates that simplify the calculations, then return the result he arrived at and translate it into the normal polar coordinates with which he started the problem. Regarding this point, Schwarzfeld's step is sound and contains intelligence and creativity that can be credited to him. This does not negate the fact that in his subsequent steps he committed something that required correction and revision, when he assumed in advance that the location of the singularity in the solution he was requesting would be at the origin point. It is now known that if he calculated it in the correct way, it would be at a different point the point which is known today as the Schwarz radius, which depends on the mass, on which the theory of black holes is based , but this is not the subject of this comment.