x���r��θr�!��^��q�}q�v��kŶ����A&)J%��MJ2���n 3 �H%R��!�i4z�������/�������~svu@��yv�ˁ���g��li&m��=>�y�tv��m�W��nst~�����;���ﴜ��n����j�z�����G�C9 cB���;��N*�}s�X�� �W X+��>� F���9{>�j�TF�v_���v���r�rM�;��.jW�z�����_,@N�ﶟ�{�+lu${1�&��3|�NN�����޻.�Vi�=�g��a�)%�u?} 5 0 obj To appear in Wei-Tou Ni (editor) \One Hundred Years of General Relativity: Cosmology and Gravity," World Scienti c, … Eventually, we present some generalizations of the Kerr solution. Schwarzschild solution is the unique spherically symmetric solution to Einstein’s equations in vacuum. Getting back to the Schwarzschild manifold, there are two obvious Killing vectors that reflect the independence of the metric on t and φ: ξα (t) = ∂t, ξ α (φ) = ∂φ. ��3�(�B��z�'��q)<6B��1�倏'Y�3#�sO� ������ ��ut� �:-d����İ mp�_�n%L �㪇͟w$(\_;je��������(RlU4�_�2(� �P@��;=()Bv$1�^b��Zi�a7諃 ׎�� � M�����ic$Ja��a84N��1@)x�E`ތ&�*�8ۻ�����ޢ��U���H�j�%.S�!�d 6H=�QU�m\���C:��(- 3���y5��M�W^�J�����|$�鴪�'�ʼ��=�G�u #��e�ց����P�����/��`C� ���Ț'h�P,R@o�e��f�K�! �cdbD/����@�^Λ�\��Tk��v@�o���7�X(b�1�!�U���*����2�k?3Ҍ狂q��l�}�����`G��� GZ�� �R�a�pǶ`Xc$��.������KO Karl Schwarzschild discovered this black hole geometry at the close of 1915 1,2,3,4,r54, within weeks of Einstein presenting his final theory of General Relativity.. 4 0 obj %PDF-1.3 The simplest kind of black hole is a Schwarzschild black hole, which is a black hole with mass, but with no electric charge, and no spin. �1H����w[��e�#.q�`����S�%�动Z�Ͳc��i>�s��9��W����x�/�!�"yU���� ����HW]+��ݡ��o1ld�qv��ց��t�ڮ�n���#�����3Uʍu"ɐAq�D0kp�(X�@���f�����-L��gD��qNr�����ޤ+��zY$��2�������a�����x�֡����P�)�§�e��������<9j;�Zx���%�+����X ΂k������E�;�I����q�?s�z�Nw���3Q7��*u,���z�����|x�Z _im�W�' Q��e0>>�v)���t��ӿ�"ꨃ�� [���� ��o69XA�����Uȥ^v��N?�p�ּ(/EB\%!���`6b���Wc�m gÃ��,���y���b���� =6L_'5� ���%�VKh��m�f��PI���j�3��6����=j�tH:�)j�摸C���hم=�5��uF�. As this metric is the correct one to use in situations within %��������� the study of black hole physics, and a great motivation for astronomers in their search for good candidates. Or “plunge” into a black hole A2290-34 Schwarzschild Metric 16 Applicability The Schwarzschild metric applies only outside the surface of an object Also, slowly spinning object like the Earth and Sun are okay A black hole has no “surface” so the Schwarzschild metric can apply to almost r = 0 (the singularity). �/L��)YV��0/D^� H�" D���T������3�3݃��]]���ʬ���Z��>V�nݶա�l����V�a�o�����������O����^s���v��6����Ѻ���v۬7���}��w@\�~�����������W���z}\_=�����z��W_^�o�v}��yj��������x�����_�o>�|[�Y&���B����Շ�4#����)������f}���@�~~&�x]���٭��]�����p�[n �h�]�? ���f}���oi��� ���O?^��뫸���9�7RC}\�C�=Tݢ*��xM��h����Z�Җ��b��~8�VVq�nS��r������3u����]XI��J��j[a'�H+��k�IB��7!�=�v��4`Du�E2W׫����2��bn ��}���=Z�3���+���nVW_]����q�s��.�+,&ZPS������^��;���B�1H:������}{:���v���7�����J.\�jC�r@���9T���x�f�! The procedure will be to first present some non-rigorous arguments that any spherically symmetric metric (whether or not it solves Einstein’s equations) must take on a 27Z�8ʳ��v[���D�2*EqS aXFYH�9�=U��;�����{@wR*rC �J���F��� ���� 7P���. This is the Schwarzschild metric. &F8����-�����tT�ϲ%��{�%L�حb˂=�W���Rzr�o��>8�ۏ�#�Fi��{ʨ <> stream �֘��@���mm��d�If@Cd��8��N�� (15) The first represents the time symmetry of the system, while the second reflects the invariance of the metric under rotations about the z-axis. S 6l�N�?�[2���*b�*�`����q��N�P�8��ۿJ��f���!d4��!�$4����~�}�X�Rܙ{�*m&+�����^��.�Iڙ ى$"�o1�W~o��́ky��t��҈Y@���,���4�'B�a5!T�JA@�7�'ڶ2�Z(���I���v"\�0�D�7{�o�C��+�K?�d�_� ���hF}*�� >$5,$&Χ:�w�셐�B& ��7����!Ӂ��*�;�v��,�����E���[XH� �����8x�1��ҡ�n�l��/��|��2-�`��JAƒ&�[�g��T���ƸCWy�Q�8n %PDF-1.4 Schwarzschild Solution and Black Holes Asaf Pe’er1 February 19, 2014 This part of the course is based on Refs. x��ے�q���)F�ڍ�6�s This equation gives us the geometry of spacetime outside of a single massive object. << /Length 5 0 R /Filter /FlateDecode >> We could use the Earth, Sun, or a black hole by inserting the appropriate mass. ?�׊�Un5g\J*=ʸ&L@݌�S�TL�g�*!��FM3�C��6n.�_E Figures 1, 10 - 14 are taken from Sean Carroll’s notes in “Level 5: A Knowledgebase for Extragalactic Astronomy ";'y�…�J�u���d�`�/�Lz�����X��8�^�xȎ�ώ�߽eG*aub0���XΉ���_��T�^�� The Schwarzschild metric . black hole is explained to some extent. le schwarzkerr29arxiv.tex, 7 March 2015 Invited review article. stream Of course, the geometry [1], [2], [3] and [4]. �����@( 1.1 Einstein’s equation The goal is to find a solution of Einstein’s equation for our metric (1), Rµν − 1 2 gµν = 8πG c4 Tµν (3) FIrst some terminology: Rµν Ricci tensor, R Ricci scalar, and Tµν stress-energy tensor (the last term will vanish for the Schwarzschild solution). Classification of black holes by type: Schwarzschild or static black hole rotating or Kerr black hole charged black hole or Newman black hole and Kerr-Newman black hole A classification of black holes by mass: micro black hole and extra-dimensional black hole primordial black hole, a hypothetical leftover of the Big Bang r*4���;�c�������1L4.k���$~Fр��������e�NU"����{�I�Jp���3]>F��$� V�{�Nz�d�/2��sd�0֠���LLj����Pg�;�^:�IH�u�r�! %�쏢 The background is Axel Mellinger's All-Sky Milky Way Panorama (by permission). \�I Ե���N�͡ ^�@ On 13 January 1916, less than two months after Einstein completed his theory of general relativity on 18 November 1915, and less than four months before his own death, the German

schwarzschild black hole pdf

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