%PDF-1.3 ^�@ stream �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�� [���� << /Length 5 0 R /Filter /FlateDecode >> %PDF-1.4 This equation gives us the geometry of spacetime outside of a single massive object. 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 27Z�8ʳ��v[���D�2*EqS aXFYH�9�=U��;�����{@wR*rC �J���F��� ���� 7P���. We could use the Earth, Sun, or a black hole by inserting the appropriate mass. \�I Ե���N�͡ Figures 1, 10 - 14 are taken from Sean Carroll’s notes in “Level 5: A Knowledgebase for Extragalactic Astronomy 5 0 obj 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. x��ے�q���)F�ڍ�6�s Schwarzschild Solution and Black Holes Asaf Pe’er1 February 19, 2014 This part of the course is based on Refs. �����@( (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. The background is Axel Mellinger's All-Sky Milky Way Panorama (by permission). stream S 6l�N�?�[2���*b�*�`����q��N�P�8��ۿJ��f���!d4��!�$4����~�}�X�Rܙ{�*m&+�����^��.�Iڙ <> le schwarzkerr29arxiv.tex, 7 March 2015 Invited review article. This is the Schwarzschild metric. Of course, the geometry ��3�(�B��z�'��q)<6B��1�倏'Y�3#�sO� ������ ��ut� �:-d����İ mp�_�n%L �㪇͟w$(\_;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�����|$�鴪�'�ʼ��=�G�u #��e�ց����P�����/��`C� ���Ț'h�P,R@o�e��f�K�! ��o69XA�����Uȥ^v��N?�p�ּ(/EB\%!���`6b���Wc�m gÃ��,���y���b���� =6L_'5� ���%�VKh��m�f��PI���j�3��6����=j�tH:�)j�摸C���hم=�5��uF�. [1], [2], [3] and [4]. ى$"�o1�W~o��́ky��t��҈Y@���,���4�'B�a5!T�JA@�7�'ڶ2�Z(���I���v"\�0�D�7{�o�C��+�K?�d�_� 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 4 0 obj ���hF}*�� >$5,$&Χ:�w�셐�B& ��7����!Ӂ��*�;�v��,�����E���[XH� �����8x�1��ҡ�n�l��/��|��2-�`��JAƒ&�[�g��T���ƸCWy�Q�8n Schwarzschild solution is the unique spherically symmetric solution to Einstein’s equations in vacuum. �cdbD/����@�^Λ�\��Tk��v@�o���7�X(b�1�!�U���*����2�k?3Ҍ狂q��l�}�����`G��� GZ�� �R�a�pǶ`Xc$��.������KO ";'y�…�J�u���d�`�/�Lz�����X��8�^�xȎ�ώ�߽eG*aub0���XΉ���_��T�^�� %��������� 1.1 Einstein’s equation The goal is to find a solution of Einstein’s equation for our metric (1), Rµν − 1 2 gµν = 8πG c4 Tµν (3) FIrst some terminology: Rµν Ricci tensor, R Ricci scalar, and Tµν stress-energy tensor (the last term will vanish for the Schwarzschild solution). black hole is explained to some extent. Or “plunge” into a black hole A2290-34 Schwarzschild Metric 16 Applicability The Schwarzschild metric applies only outside the surface of an object Also, slowly spinning object like the Earth and Sun are okay A black hole has no “surface” so the Schwarzschild metric can apply to almost r = 0 (the singularity). To appear in Wei-Tou Ni (editor) \One Hundred Years of General Relativity: Cosmology and Gravity," World Scienti c, … &F8����-�����tT�ϲ%��{�%L�حb˂=�W���Rzr�o��>8�ۏ�#�Fi��{ʨ ?�׊�Un5g\J*=ʸ&L@݌�S�TL�g�*!��FM3�C��6n.�_E the study of black hole physics, and a great motivation for astronomers in their search for good candidates. The Schwarzschild metric . Karl Schwarzschild discovered this black hole geometry at the close of 1915 1,2,3,4,r54, within weeks of Einstein presenting his final theory of General Relativity.. �֘��@���mm��d�If@Cd��8��N�� Eventually, we present some generalizations of the Kerr solution. The simplest kind of black hole is a Schwarzschild black hole, which is a black hole with mass, but with no electric charge, and no spin. x���r��θr�!��^��q�}q�v��kŶ����A&)J%��MJ2���n 3 �H%R��!�i4z�������/�������~svu@��yv�ˁ���g��li&m��=>�y�tv��m�W��nst~�����;���ﴜ��n����j�z�����G�C9 cB���;��N*�}s�X�� �W X+��>� F���9{>�j�TF�v_���v���r�rM�;��.jW�z�����_,@N�ﶟ�{�+lu${1�&��3|�NN�����޻.�Vi�=�g��a�)%�u?} 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 r*4���;�c�������1L4.k���$~Fр��������e�NU"����{�I�Jp���3]>F��$� V�{�Nz�d�/2��sd�0֠���LLj����Pg�;�^:�IH�u�r�! %�쏢 �/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�! As this metric is the correct one to use in situations within