Post on 24-Jan-2021
Take background to be Schwarzschild, consider vacuum perturbations:
Build Ricci perturbation from hαβ; break hαβ into scalar, vector, tensor pieces;
characterize them by their parity properties.
Perturbing Schwarzschildds2 =
�gB↵� + h↵�
�dx↵dx�
<latexit sha1_base64="Gpw2cFWpW8oAqC/iE+UOHBHtub8=">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</latexit>
G↵� = 0 �! �R↵� = 0<latexit sha1_base64="xrBIe0AkIs18AVJ4Z5KbAp06i+8=">AAACJ3icbVBNaxsxFNSmTZO6abNpjr2ImkJPZtcNJJcEkxzSYxriD/Aa81Z+tkW00iK9TTCL/00u+Su5BNJQ2mP/SeWPQ2t3QDDMzOPpTZor6SiKfgUbL15uvtrafl15s/P23W64977lTGEFNoVRxnZScKikxiZJUtjJLUKWKmyn12czv32D1kmjr2iSYy+DkZZDKYC81A9PzvtlAiofQ5IiwZQf84gnyuiRlaMxgbXmlicDVAT8cj3aD6tRLZqDr5N4SapsiYt++JQMjCgy1CQUONeNo5x6JViSQuG0khQOcxDXMMKupxoydL1yfueUf/LKgA+N9U8Tn6t/T5SQOTfJUp/MgMZu1ZuJ//O6BQ2PeqXUeUGoxWLRsFCcDJ+VxgfSoiA18QSElf6vXIzBgiBfbcWXEK+evE5a9Vr8pVb/dlBtnC7r2GYf2Ef2mcXskDXYV3bBmkywO/bAvrPn4D54DH4EPxfRjWA5s8/+QfD7D0REpYs=</latexit>
See Rezzolla, gr-qc/0302025 for details and the explicit form of the basis tensors (which
include first and second derivatives of Ylm).
Can put m = 0 (axisymmetry); can set H2 = 0 by choice of gauge.
Results for odd parityh00 = 0
<latexit sha1_base64="TbdweOue9gGwja8KDj/l3ruRg+k=">AAAB8XicbVBNSwMxEJ34WetX1aOXYBE8lWwV9CIUvXisYD+wXUo2zbah2eySZIWy9F948aCIV/+NN/+NabsHbX0w8Hhvhpl5QSKFsYR8o5XVtfWNzcJWcXtnd2+/dHDYNHGqGW+wWMa6HVDDpVC8YYWVvJ1oTqNA8lYwup36rSeujYjVgx0n3I/oQIlQMGqd9DjsZYRM8DUmvVKZVMgMeJl4OSlDjnqv9NXtxyyNuLJMUmM6Hkmsn1FtBZN8UuymhieUjeiAdxxVNOLGz2YXT/CpU/o4jLUrZfFM/T2R0ciYcRS4zojaoVn0puJ/Xie14ZWfCZWklis2XxSmEtsYT9/HfaE5s3LsCGVauFsxG1JNmXUhFV0I3uLLy6RZrXjnler9Rbl2k8dRgGM4gTPw4BJqcAd1aAADBc/wCm/IoBf0jj7mrSsonzmCP0CfP8ppj64=</latexit>
h0i.= H0(t, r) [0,� csc ✓@�Y`m, sin ✓@✓Y`m]
<latexit sha1_base64="Z+FYHbpL7O2PQqRGtOhYskhvEuY=">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</latexit>
hij = H1(t, r)e1ij +H2(t, r)e
2ij
<latexit sha1_base64="JIAMgzdvHlcCpUbBa1a/XQy7ndY=">AAACJ3icbZDLSsNAFIYn9VbrLerSzWARKkpJoqAbpeimywr2Am0Nk+mkHTu5MDMRSsjbuPFV3AgqokvfxEmahbYeGPj5v3OYc34nZFRIw/jSCguLS8srxdXS2vrG5pa+vdMSQcQxaeKABbzjIEEY9UlTUslIJ+QEeQ4jbWd8nfL2A+GCBv6tnISk76GhT12KkVSWrV+O7JjeJ/AC1m2zIo/5Yc9DcuS4MUnuzCk7UsyaZVbGbL1sVI2s4Lwwc1EGeTVs/bU3CHDkEV9ihoTomkYo+zHikmJGklIvEiREeIyGpKukjzwi+nF2ZwIPlDOAbsDV8yXM3N8TMfKEmHiO6kwXFbMsNf9j3Ui65/2Y+mEkiY+nH7kRgzKAaWhwQDnBkk2UQJhTtSvEI8QRlirakgrBnD15XrSsqnlStW5Oy7WrPI4i2AP7oAJMcAZqoA4aoAkweATP4A28a0/ai/ahfU5bC1o+swv+lPb9A/vzpNU=</latexit>
As r ➞ ∞ and r ➞ 2GM, this simplifies:
The Regge-Wheeler equation@2Q
@t2� @2Q
@r2⇤+
✓1� 2GM
r
◆`(`+ 1)
r2� 6GM
r3
�Q = 0
<latexit sha1_base64="3WFCmqcOaJ5oEunoZLw42HsWPIU=">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</latexit>
r⇤ = r + 2GM ln⇣ r
2GM� 1
⌘
<latexit sha1_base64="Fx/PmQYsF9m51NK9phXFaTsPUmc=">AAACF3icbVDLSgNBEJz1GeNr1aOXwSBExbAbBb0IQQ96ESKYB2RDmJ3MJkNmZ5eZXiEs+Qsv/ooXD4p41Zt/4+Rx0MSChqKqm+4uPxZcg+N8W3PzC4tLy5mV7Ora+samvbVd1VGiKKvQSESq7hPNBJesAhwEq8eKkdAXrOb3roZ+7YEpzSN5D/2YNUPSkTzglICRWnZBtQ7xBVb4CBevbz0hPcECyHuBIjRVg9SIA3yMXU/xThcOWnbOKTgj4FniTkgOTVBu2V9eO6JJyCRQQbRuuE4MzZQo4FSwQdZLNIsJ7ZEOaxgqSch0Mx39NcD7RmnjIFKmJOCR+nsiJaHW/dA3nSGBrp72huJ/XiOB4LyZchknwCQdLwoSgSHCw5BwmytGQfQNIVRxcyumXWIiARNl1oTgTr88S6rFgntSKN6d5kqXkzgyaBftoTxy0RkqoRtURhVE0SN6Rq/ozXqyXqx362PcOmdNZnbQH1ifP/9+nVw=</latexit>
Q ⌘ H1
r
✓1� 2GM
r
◆
<latexit sha1_base64="GGnq1rHkCUYwsPLGbWsoTWiHEjM=">AAACHXicbZA9SwNBEIb3/Izx69TSZjEIsTDcxYCWooVphASMCrkQ9jZzyeLeh7tzQjjyR2z8KzYWiljYiP/GTbxCoy8svDwzw+y8fiKFRsf5tGZm5+YXFgtLxeWV1bV1e2PzUsep4tDisYzVtc80SBFBCwVKuE4UsNCXcOXfnI7rV3egtIijCxwm0AlZPxKB4AwN6tq1JvXgNhV31AsU41m9644yNfIkBFh26X6Oq2fnE6xEf4B7XbvkVJyJ6F/j5qZEcjW69rvXi3kaQoRcMq3brpNgJ2MKBZcwKnqphoTxG9aHtrERC0F3ssl1I7prSI8GsTIvQjqhPycyFmo9DH3TGTIc6OnaGP5Xa6cYHHUyESUpQsS/FwWppBjTcVS0JxRwlENjGFfC/JXyATNxoAm0aEJwp0/+ay6rFfegUm3WSscneRwFsk12SJm45JAckzppkBbh5J48kmfyYj1YT9ar9fbdOmPlM1vkl6yPL4DmoZA=</latexit>
@H0
@t=
@
@r⇤(r⇤Q)
<latexit sha1_base64="6IFjtg0jcYgAPjwL4yaQMW+Cuhk=">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</latexit>
@2Q
@t2� @2Q
@r2⇤= 0
<latexit sha1_base64="8tn0hLBsAPqRAAIgcVBZG6FCtQk=">AAACM3icbVDLSgMxFM3UV62vUZdugkUQwTIzCroRim7EVQv2AZ3pkEkzbWjmQZIRyjD/5MYfcSGIC0Xc+g9m2iK19UDg5Nxzb3KPFzMqpGG8aoWl5ZXVteJ6aWNza3tH391riijhmDRwxCLe9pAgjIakIalkpB1zggKPkZY3vMnrrQfCBY3CezmKiROgfkh9ipFUkqvf2T5HOLVjxCVFrGvBevZ7g7JrZfAUzntmLdw9yU1X0HD1slExxoCLxJySMpii5urPdi/CSUBCiRkSomMasXTSfDBmJCvZiSAxwkPUJx1FQxQQ4aTjnTN4pJQe9COuTijhWJ3tSFEgxCjwlDNAciDma7n4X62TSP/SSWkYJ5KEePKQn6gsIpgHCHuUEyzZSBGEOVV/hXiAVEBSxVxSIZjzKy+SplUxzypW/bxcvZ7GUQQH4BAcAxNcgCq4BTXQABg8ghfwDj60J+1N+9S+JtaCNu3ZB3+gff8AzraqUg==</latexit>
Q ⇠ ei!(t±r⇤)<latexit sha1_base64="7QX+WmXscFBF6P0m2xZHvFDkUMQ=">AAACB3icbVDLSsNAFJ3UV62vqEtBBotQXZSkCrosunHZgn1AE8NketsOnUnCzEQooTs3/oobF4q49Rfc+TdOHwutHrhwOOde7r0nTDhT2nG+rNzS8srqWn69sLG5tb1j7+41VZxKCg0a81i2Q6KAswgammkO7UQCESGHVji8nvite5CKxdGtHiXgC9KPWI9Roo0U2Id17CkmMNxlzIsF9AkuaewlAsvg9GQc2EWn7EyB/xJ3Topojlpgf3rdmKYCIk05UarjOon2MyI1oxzGBS9VkBA6JH3oGBoRAcrPpn+M8bFRurgXS1ORxlP150RGhFIjEZpOQfRALXoT8T+vk+repZ+xKEk1RHS2qJdyrGM8CQV3mQSq+cgQQiUzt2I6IJJQbaIrmBDcxZf/kmal7J6VK/XzYvVqHkceHaAjVEIuukBVdINqqIEoekBP6AW9Wo/Ws/Vmvc9ac9Z8Zh/9gvXxDb9tl/M=</latexit>
so in this limit.
Minus sign corresponds to outgoing wave packet; plus sign to ingoing.
As r ➞ ∞ and r ➞ 2GM, this simplifies:
The Regge-Wheeler equation@2Q
@t2� @2Q
@r2⇤+
✓1� 2GM
r
◆`(`+ 1)
r2� 6GM
r3
�Q = 0
<latexit sha1_base64="3WFCmqcOaJ5oEunoZLw42HsWPIU=">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</latexit>
r⇤ = r + 2GM ln⇣ r
2GM� 1
⌘
<latexit sha1_base64="Fx/PmQYsF9m51NK9phXFaTsPUmc=">AAACF3icbVDLSgNBEJz1GeNr1aOXwSBExbAbBb0IQQ96ESKYB2RDmJ3MJkNmZ5eZXiEs+Qsv/ooXD4p41Zt/4+Rx0MSChqKqm+4uPxZcg+N8W3PzC4tLy5mV7Ora+samvbVd1VGiKKvQSESq7hPNBJesAhwEq8eKkdAXrOb3roZ+7YEpzSN5D/2YNUPSkTzglICRWnZBtQ7xBVb4CBevbz0hPcECyHuBIjRVg9SIA3yMXU/xThcOWnbOKTgj4FniTkgOTVBu2V9eO6JJyCRQQbRuuE4MzZQo4FSwQdZLNIsJ7ZEOaxgqSch0Mx39NcD7RmnjIFKmJOCR+nsiJaHW/dA3nSGBrp72huJ/XiOB4LyZchknwCQdLwoSgSHCw5BwmytGQfQNIVRxcyumXWIiARNl1oTgTr88S6rFgntSKN6d5kqXkzgyaBftoTxy0RkqoRtURhVE0SN6Rq/ozXqyXqx362PcOmdNZnbQH1ifP/9+nVw=</latexit>
Q ⌘ H1
r
✓1� 2GM
r
◆
<latexit sha1_base64="GGnq1rHkCUYwsPLGbWsoTWiHEjM=">AAACHXicbZA9SwNBEIb3/Izx69TSZjEIsTDcxYCWooVphASMCrkQ9jZzyeLeh7tzQjjyR2z8KzYWiljYiP/GTbxCoy8svDwzw+y8fiKFRsf5tGZm5+YXFgtLxeWV1bV1e2PzUsep4tDisYzVtc80SBFBCwVKuE4UsNCXcOXfnI7rV3egtIijCxwm0AlZPxKB4AwN6tq1JvXgNhV31AsU41m9644yNfIkBFh26X6Oq2fnE6xEf4B7XbvkVJyJ6F/j5qZEcjW69rvXi3kaQoRcMq3brpNgJ2MKBZcwKnqphoTxG9aHtrERC0F3ssl1I7prSI8GsTIvQjqhPycyFmo9DH3TGTIc6OnaGP5Xa6cYHHUyESUpQsS/FwWppBjTcVS0JxRwlENjGFfC/JXyATNxoAm0aEJwp0/+ay6rFfegUm3WSscneRwFsk12SJm45JAckzppkBbh5J48kmfyYj1YT9ar9fbdOmPlM1vkl6yPL4DmoZA=</latexit>
@H0
@t=
@
@r⇤(r⇤Q)
<latexit sha1_base64="6IFjtg0jcYgAPjwL4yaQMW+Cuhk=">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</latexit>
@2Q
@t2� @2Q
@r2⇤= 0
<latexit sha1_base64="8tn0hLBsAPqRAAIgcVBZG6FCtQk=">AAACM3icbVDLSgMxFM3UV62vUZdugkUQwTIzCroRim7EVQv2AZ3pkEkzbWjmQZIRyjD/5MYfcSGIC0Xc+g9m2iK19UDg5Nxzb3KPFzMqpGG8aoWl5ZXVteJ6aWNza3tH391riijhmDRwxCLe9pAgjIakIalkpB1zggKPkZY3vMnrrQfCBY3CezmKiROgfkh9ipFUkqvf2T5HOLVjxCVFrGvBevZ7g7JrZfAUzntmLdw9yU1X0HD1slExxoCLxJySMpii5urPdi/CSUBCiRkSomMasXTSfDBmJCvZiSAxwkPUJx1FQxQQ4aTjnTN4pJQe9COuTijhWJ3tSFEgxCjwlDNAciDma7n4X62TSP/SSWkYJ5KEePKQn6gsIpgHCHuUEyzZSBGEOVV/hXiAVEBSxVxSIZjzKy+SplUxzypW/bxcvZ7GUQQH4BAcAxNcgCq4BTXQABg8ghfwDj60J+1N+9S+JtaCNu3ZB3+gff8AzraqUg==</latexit>
Q ⇠ ei!(t±r⇤)<latexit sha1_base64="7QX+WmXscFBF6P0m2xZHvFDkUMQ=">AAACB3icbVDLSsNAFJ3UV62vqEtBBotQXZSkCrosunHZgn1AE8NketsOnUnCzEQooTs3/oobF4q49Rfc+TdOHwutHrhwOOde7r0nTDhT2nG+rNzS8srqWn69sLG5tb1j7+41VZxKCg0a81i2Q6KAswgammkO7UQCESGHVji8nvite5CKxdGtHiXgC9KPWI9Roo0U2Id17CkmMNxlzIsF9AkuaewlAsvg9GQc2EWn7EyB/xJ3Topojlpgf3rdmKYCIk05UarjOon2MyI1oxzGBS9VkBA6JH3oGBoRAcrPpn+M8bFRurgXS1ORxlP150RGhFIjEZpOQfRALXoT8T+vk+repZ+xKEk1RHS2qJdyrGM8CQV3mQSq+cgQQiUzt2I6IJJQbaIrmBDcxZf/kmal7J6VK/XzYvVqHkceHaAjVEIuukBVdINqqIEoekBP6AW9Wo/Ws/Vmvc9ac9Z8Zh/9gvXxDb9tl/M=</latexit>
so in this limit.
Physics: Nothing can come out of the event horizon; nothing can come in from infinity.
The Regge-Wheeler equation@2Q
@t2� @2Q
@r2⇤+
✓1� 2GM
r
◆`(`+ 1)
r2� 6GM
r3
�Q = 0
<latexit sha1_base64="3WFCmqcOaJ5oEunoZLw42HsWPIU=">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</latexit>
r⇤ = r + 2GM ln⇣ r
2GM� 1
⌘
<latexit sha1_base64="Fx/PmQYsF9m51NK9phXFaTsPUmc=">AAACF3icbVDLSgNBEJz1GeNr1aOXwSBExbAbBb0IQQ96ESKYB2RDmJ3MJkNmZ5eZXiEs+Qsv/ooXD4p41Zt/4+Rx0MSChqKqm+4uPxZcg+N8W3PzC4tLy5mV7Ora+samvbVd1VGiKKvQSESq7hPNBJesAhwEq8eKkdAXrOb3roZ+7YEpzSN5D/2YNUPSkTzglICRWnZBtQ7xBVb4CBevbz0hPcECyHuBIjRVg9SIA3yMXU/xThcOWnbOKTgj4FniTkgOTVBu2V9eO6JJyCRQQbRuuE4MzZQo4FSwQdZLNIsJ7ZEOaxgqSch0Mx39NcD7RmnjIFKmJOCR+nsiJaHW/dA3nSGBrp72huJ/XiOB4LyZchknwCQdLwoSgSHCw5BwmytGQfQNIVRxcyumXWIiARNl1oTgTr88S6rFgntSKN6d5kqXkzgyaBftoTxy0RkqoRtURhVE0SN6Rq/ozXqyXqx362PcOmdNZnbQH1ifP/9+nVw=</latexit>
Q ⌘ H1
r
✓1� 2GM
r
◆
<latexit sha1_base64="GGnq1rHkCUYwsPLGbWsoTWiHEjM=">AAACHXicbZA9SwNBEIb3/Izx69TSZjEIsTDcxYCWooVphASMCrkQ9jZzyeLeh7tzQjjyR2z8KzYWiljYiP/GTbxCoy8svDwzw+y8fiKFRsf5tGZm5+YXFgtLxeWV1bV1e2PzUsep4tDisYzVtc80SBFBCwVKuE4UsNCXcOXfnI7rV3egtIijCxwm0AlZPxKB4AwN6tq1JvXgNhV31AsU41m9644yNfIkBFh26X6Oq2fnE6xEf4B7XbvkVJyJ6F/j5qZEcjW69rvXi3kaQoRcMq3brpNgJ2MKBZcwKnqphoTxG9aHtrERC0F3ssl1I7prSI8GsTIvQjqhPycyFmo9DH3TGTIc6OnaGP5Xa6cYHHUyESUpQsS/FwWppBjTcVS0JxRwlENjGFfC/JXyATNxoAm0aEJwp0/+ay6rFfegUm3WSscneRwFsk12SJm45JAckzppkBbh5J48kmfyYj1YT9ar9fbdOmPlM1vkl6yPL4DmoZA=</latexit>
@H0
@t=
@
@r⇤(r⇤Q)
<latexit sha1_base64="6IFjtg0jcYgAPjwL4yaQMW+Cuhk=">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</latexit>
Require Q ⇠ ei!(t�r⇤) r ! 1<latexit sha1_base64="G+ZLASOU6r5vyavLCdPZIVtFPmU=">AAACF3icbVDLSgMxFM34tr6qLt0Ei6CCZaYKuiy6calgW6FTSya9U4NJZprcEcrQv3Djr7hxoYhb3fk3prULXwcCh3Pu5eacKJXCou9/eBOTU9Mzs3PzhYXFpeWV4upa3SaZ4VDjiUzMZcQsSKGhhgIlXKYGmIokNKKbk6HfuAVjRaIvsJ9CS7GuFrHgDJ3ULpbPaWiFonCVizBR0GXbSPeoae/uDMJeL2MdamiICQ2FjrHfLpb8sj8C/UuCMSmRMc7axfewk/BMgUYumbXNwE+xlTODgksYFMLMQsr4DetC01HNFNhWPso1oFtO6dA4Me5ppCP1+0bOlLV9FblJxfDa/vaG4n9eM8P4qJULnWYImn8dijNJXc5hSbQjDHCUfUcYN8L9lfJrZhhHV2XBlRD8jvyX1CvlYL9cOT8oVY/HdcyRDbJJtklADkmVnJIzUiOc3JEH8kSevXvv0XvxXr9GJ7zxzjr5Ae/tE8vOnnY=</latexit>
Q ⇠ ei!(t+r⇤) r ! 2GM<latexit sha1_base64="gTJa7u5cWpYnst1aKo+CoTalj2s=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUC2WmCrosutCN0IJ9QKeWTHrbhiYz0yQjlKEf4cZfceNCEbcu3Pk3po+Fth4IHM65l5tzvJAzpW3720osLa+sriXXUxubW9s76d29qgoiSaFCAx7IukcUcOZDRTPNoR5KIMLjUPP6V2O/9gBSscC/08MQmoJ0fdZhlGgjtdK5MnYVExjuY+YGArokq3EOy9bJ8cgdDCLSxhK7OsCF69tWOmPn7QnwInFmJINmKLXSX247oJEAX1NOlGo4dqibMZGaUQ6jlBspCAntky40DPWJANWMJ6FG+MgobdwJpHm+xhP190ZMhFJD4ZlJQXRPzXtj8T+vEenORTNmfhhp8On0UCfi2IQcN4TbTALVfGgIoZKZv2LaI5JQbXpMmRKc+ciLpFrIO6f5QvksU7yc1ZFEB+gQZZGDzlER3aASqiCKHtEzekVv1pP1Yr1bH9PRhDXb2Ud/YH3+AIO0nJY=</latexit>
The Regge-Wheeler equation@2Q
@t2� @2Q
@r2⇤+
✓1� 2GM
r
◆`(`+ 1)
r2� 6GM
r3
�Q = 0
<latexit sha1_base64="3WFCmqcOaJ5oEunoZLw42HsWPIU=">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</latexit>
r⇤ = r + 2GM ln⇣ r
2GM� 1
⌘
<latexit sha1_base64="Fx/PmQYsF9m51NK9phXFaTsPUmc=">AAACF3icbVDLSgNBEJz1GeNr1aOXwSBExbAbBb0IQQ96ESKYB2RDmJ3MJkNmZ5eZXiEs+Qsv/ooXD4p41Zt/4+Rx0MSChqKqm+4uPxZcg+N8W3PzC4tLy5mV7Ora+samvbVd1VGiKKvQSESq7hPNBJesAhwEq8eKkdAXrOb3roZ+7YEpzSN5D/2YNUPSkTzglICRWnZBtQ7xBVb4CBevbz0hPcECyHuBIjRVg9SIA3yMXU/xThcOWnbOKTgj4FniTkgOTVBu2V9eO6JJyCRQQbRuuE4MzZQo4FSwQdZLNIsJ7ZEOaxgqSch0Mx39NcD7RmnjIFKmJOCR+nsiJaHW/dA3nSGBrp72huJ/XiOB4LyZchknwCQdLwoSgSHCw5BwmytGQfQNIVRxcyumXWIiARNl1oTgTr88S6rFgntSKN6d5kqXkzgyaBftoTxy0RkqoRtURhVE0SN6Rq/ozXqyXqx362PcOmdNZnbQH1ifP/9+nVw=</latexit>
Q ⌘ H1
r
✓1� 2GM
r
◆
<latexit sha1_base64="GGnq1rHkCUYwsPLGbWsoTWiHEjM=">AAACHXicbZA9SwNBEIb3/Izx69TSZjEIsTDcxYCWooVphASMCrkQ9jZzyeLeh7tzQjjyR2z8KzYWiljYiP/GTbxCoy8svDwzw+y8fiKFRsf5tGZm5+YXFgtLxeWV1bV1e2PzUsep4tDisYzVtc80SBFBCwVKuE4UsNCXcOXfnI7rV3egtIijCxwm0AlZPxKB4AwN6tq1JvXgNhV31AsU41m9644yNfIkBFh26X6Oq2fnE6xEf4B7XbvkVJyJ6F/j5qZEcjW69rvXi3kaQoRcMq3brpNgJ2MKBZcwKnqphoTxG9aHtrERC0F3ssl1I7prSI8GsTIvQjqhPycyFmo9DH3TGTIc6OnaGP5Xa6cYHHUyESUpQsS/FwWppBjTcVS0JxRwlENjGFfC/JXyATNxoAm0aEJwp0/+ay6rFfegUm3WSscneRwFsk12SJm45JAckzppkBbh5J48kmfyYj1YT9ar9fbdOmPlM1vkl6yPL4DmoZA=</latexit>
@H0
@t=
@
@r⇤(r⇤Q)
<latexit sha1_base64="6IFjtg0jcYgAPjwL4yaQMW+Cuhk=">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</latexit>
Require Q ⇠ ei!(t�r⇤) r ! 1<latexit sha1_base64="G+ZLASOU6r5vyavLCdPZIVtFPmU=">AAACF3icbVDLSgMxFM34tr6qLt0Ei6CCZaYKuiy6calgW6FTSya9U4NJZprcEcrQv3Djr7hxoYhb3fk3prULXwcCh3Pu5eacKJXCou9/eBOTU9Mzs3PzhYXFpeWV4upa3SaZ4VDjiUzMZcQsSKGhhgIlXKYGmIokNKKbk6HfuAVjRaIvsJ9CS7GuFrHgDJ3ULpbPaWiFonCVizBR0GXbSPeoae/uDMJeL2MdamiICQ2FjrHfLpb8sj8C/UuCMSmRMc7axfewk/BMgUYumbXNwE+xlTODgksYFMLMQsr4DetC01HNFNhWPso1oFtO6dA4Me5ppCP1+0bOlLV9FblJxfDa/vaG4n9eM8P4qJULnWYImn8dijNJXc5hSbQjDHCUfUcYN8L9lfJrZhhHV2XBlRD8jvyX1CvlYL9cOT8oVY/HdcyRDbJJtklADkmVnJIzUiOc3JEH8kSevXvv0XvxXr9GJ7zxzjr5Ae/tE8vOnnY=</latexit>
Q ⇠ ei!(t+r⇤) r ! 2GM<latexit sha1_base64="gTJa7u5cWpYnst1aKo+CoTalj2s=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUC2WmCrosutCN0IJ9QKeWTHrbhiYz0yQjlKEf4cZfceNCEbcu3Pk3po+Fth4IHM65l5tzvJAzpW3720osLa+sriXXUxubW9s76d29qgoiSaFCAx7IukcUcOZDRTPNoR5KIMLjUPP6V2O/9gBSscC/08MQmoJ0fdZhlGgjtdK5MnYVExjuY+YGArokq3EOy9bJ8cgdDCLSxhK7OsCF69tWOmPn7QnwInFmJINmKLXSX247oJEAX1NOlGo4dqibMZGaUQ6jlBspCAntky40DPWJANWMJ6FG+MgobdwJpHm+xhP190ZMhFJD4ZlJQXRPzXtj8T+vEenORTNmfhhp8On0UCfi2IQcN4TbTALVfGgIoZKZv2LaI5JQbXpMmRKc+ciLpFrIO6f5QvksU7yc1ZFEB+gQZZGDzlER3aASqiCKHtEzekVv1pP1Yr1bH9PRhDXb2Ud/YH3+AIO0nJY=</latexit>
Turns into an eigenvalue problem: There exist special frequencies such that Q is purely ingoing
on the horizon, purely outgoing at infinity.
Frequencies which do this depend on the angular index l, and have real and imaginary parts:
Quasi-normal modes
Outgoing waves then have the form
! = !r + i!i<latexit sha1_base64="mN6u1aqXXnSqjkFasWsxuQtIfac=">AAACB3icbZDLSgMxFIYzXmu9jboUJFgEQSgzVdCNUHTjsoK9QDsMmTTThuYyJBmhDN258VXcuFDEra/gzrcxbUfQ1h8CX/5zDsn5o4RRbTzvy1lYXFpeWS2sFdc3Nre23Z3dhpapwqSOJZOqFSFNGBWkbqhhpJUognjESDMaXI/rzXuiNJXizgwTEnDUEzSmGBlrhe5BR3LSQ/ASTiFU8ATSnwsN3ZJX9iaC8+DnUAK5aqH72elKnHIiDGZI67bvJSbIkDIUMzIqdlJNEoQHqEfaFgXiRAfZZI8RPLJOF8ZS2SMMnLi/JzLEtR7yyHZyZPp6tjY2/6u1UxNfBBkVSWqIwNOH4pRBI+E4FNilimDDhhYQVtT+FeI+UggbG13RhuDPrjwPjUrZPy1Xbs9K1as8jgLYB4fgGPjgHFTBDaiBOsDgATyBF/DqPDrPzpvzPm1dcPKZPfBHzsc3702YFA==</latexit>
Q ⇠ e�t/⌧ei!r(t�r⇤) , ⌧ = 1/!i<latexit sha1_base64="xCBGAIxQZdRrvqFL7I+gSOS4JnM=">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</latexit>
Values of ωr and τ in general need to be found numerically … l = 2 yields the longest lived modes:
!r ' 0.37
GM! fr ⌘ !r
2⇡= 240Hz
✓50M�M
◆
<latexit sha1_base64="gYvMKpdLikTUR+wILko2ouh0sr0=">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</latexit>
⌧!r ' 4<latexit sha1_base64="bMhjnnaRy/4zEoVHZpuS1kwX7cs=">AAAB/nicbVBNS8NAEN34WetXVDx5WSyCp5LUgh6LXjxWsB/QhLDZTtqlu0nc3QglFPwrXjwo4tXf4c1/47bNQVsfDDzem2FmXphyprTjfFsrq2vrG5ulrfL2zu7evn1w2FZJJim0aMIT2Q2JAs5iaGmmOXRTCUSEHDrh6Gbqdx5BKpbE93qcgi/IIGYRo0QbKbCPPU0yLxEwIIHEnmICHnA9sCtO1ZkBLxO3IBVUoBnYX14/oZmAWFNOlOq5Tqr9nEjNKIdJ2csUpISOyAB6hsZEgPLz2fkTfGaUPo4SaSrWeKb+nsiJUGosQtMpiB6qRW8q/uf1Mh1d+TmL00xDTOeLooxjneBpFrjPJFDNx4YQKpm5FdMhkYRqk1jZhOAuvrxM2rWqe1Gt3dUrjesijhI6QafoHLnoEjXQLWqiFqIoR8/oFb1ZT9aL9W59zFtXrGLmCP2B9fkDsImVTA==</latexit>
Metric expansion doesn’t work as well for Kerr since we no longer have spherical symmetry.
Kerr
However, a minor miracle occurs if you perturb the curvature: Recall pset exercise in which you used
Bianchi to make a wave equation for Riemann:
The first term is a wave operator; applying identities becomes a wave equation with “Riemann squared”
acting as a source term. Split Riemann:
r� [r�R↵��✏ +r↵R���✏ +r�R�↵�✏] = 0<latexit sha1_base64="4BZsr8peQ4WUI78Gj8GRH7irnIc=">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</latexit>
R↵��� = RB↵��� + �R↵���
<latexit sha1_base64="ILIIKtGVvaabTbyoMctCIfylbS0=">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</latexit>
In vacuum, Ricci curvature vanishes; Riemann is equivalent to Weyl: Rαβγδ ➞ Cαβγδ.
Vacuum; pick components
Can also organize the components by projecting them onto a special set of basis vectors:
(Tangent to ingoing null geodesics)l↵.=
1
�
⇥(r2 + a2),�, 0, a
⇤
n↵ .=
1
2(r2 + a2 cos2 ✓)
⇥(r2 + a2),��, 0, a
⇤
m↵ .=
1p2(r + ia cos ✓)
[ia sin ✓, 0, 1, i csc ✓]<latexit sha1_base64="pc0UYt20vTdFqdmGuzCTpu2l07Q=">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</latexit>
(Tangent to outgoing null)
(This plus complex conjugate cover the angular degrees of freedom)
10 degrees of freedom in the Weyl curvature are given by the following 5 complex numbers:
Complex Weyl scalars
Introduce perturbative expansion for these quantities:
0 = �C↵���l↵m�l�m�
<latexit sha1_base64="zx3Iu7BtjnwTHGjpa6lR31cXGfM=">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</latexit>
4 = �C↵���n↵m̄�n�m̄�
<latexit sha1_base64="tbxCimG31jikJckoL22bRI7sHNc=">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</latexit>
1 = �C↵���l↵n�l�m�
<latexit sha1_base64="s3bUERFxlzWjZqla+DumuTyrS9Y=">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</latexit>
3 = �C↵���l↵n�m̄�n�
<latexit sha1_base64="iZdWeskIBeXUBz8alseQsgiE3ZA=">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</latexit>
2 = �C↵���l↵m�m̄�n�
<latexit sha1_base64="KZm7ZJtOzRXe32+rRD2wKuflHB8=">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</latexit>
n = Bn + � n
<latexit sha1_base64="E5HPndYy6LmuZFw97qRX60hHXt8=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRBKEkVdCNUOrGZQV7gSaGyWTSDp1MwsxEKKFv4MZXceNCEbdu3fk2TpsstPWHgY//nMOZ8/sJo1JZ1rdRWlpeWV0rr1c2Nre2d8zdvY6MU4FJG8csFj0fScIoJ21FFSO9RBAU+Yx0/dH1tN59IELSmN+pcULcCA04DSlGSlueeey0JPU4vII53GeOiGBzAk+hExCmUG57ZtWqWTPBRbALqIJCLc/8coIYpxHhCjMkZd+2EuVmSCiKGZlUnFSSBOERGpC+Ro4iIt1sds8EHmkngGEs9OMKztzfExmKpBxHvu6MkBrK+drU/K/WT1V46WaUJ6kiHOeLwpRBFcNpODCggmDFxhoQFlT/FeIhEggrHWFFh2DPn7wInXrNPqvVb8+rjWYRRxkcgENwAmxwARrgBrRAG2DwCJ7BK3gznowX4934yFtLRjGzD/7I+PwBDUWbaw==</latexit>
Remake the Riemann wave equation in terms of Done by Teukolsky in 1973.
� n<latexit sha1_base64="zZGdFBa+G7Q38xrtpERRZJCVFU8=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YAmlM1m0i7dbMLuRiilf8OLB0W8+me8+W/ctjlo64OBx3szzMwLM8G1cd1vZ219Y3Nru7RT3t3bPzisHB23dZorhi2WilR1Q6pRcIktw43AbqaQJqHATji6m/mdJ1Sap/LRjDMMEjqQPOaMGiv5foTCUL+peV/2K1W35s5BVolXkCoUaPYrX36UsjxBaZigWvc8NzPBhCrDmcBp2c81ZpSN6AB7lkqaoA4m85un5NwqEYlTZUsaMld/T0xoovU4CW1nQs1QL3sz8T+vl5v4JphwmeUGJVssinNBTEpmAZCIK2RGjC2hTHF7K2FDqigzNqayDcFbfnmVtOs177JWf7iqNm6LOEpwCmdwAR5cQwPuoQktYJDBM7zCm5M7L86787FoXXOKmRP4A+fzBw2lkbE=</latexit>
Focus on Ψ4. The resulting equation describes radiation far from the perturbed black hole. The
equation that results turns out to separate:
Result 1: Ringing modes with spin
Fairly simple equations govern the behavior of R and S.
Again find special frequencies for which Ψ4 is purely ingoing boundary at horizon, outgoing at infinity:
4 =1
(r � ia cos ✓)4
Zd!
X
`m
R`m!(r)S`m!(✓)eim�e�i!t
<latexit sha1_base64="YGcm8JbCgPz79X2H1df3a27hkRY=">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</latexit>
4(r ! 1) ⇠ e�t/⌧`mei!`mt<latexit sha1_base64="u0LTk14x7+RAUd7awtw9NxXNOfg=">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</latexit>
!22 ⇡ 1
GM
⇥1� 0.63(1� a)0.7
⇤<latexit sha1_base64="DNijgmFoTrfWZlhCYq7+PdcWQWQ=">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</latexit>
⌧22!22 ⇡ 4(1� a)�0.45<latexit sha1_base64="xlA9COPOGALv5Qht4KV3pW3F8Gg=">AAACF3icbZDLTgIxFIY7XhFvqEs3jcQEF0xmRowuiW5cYiKXBJCcKQUaOtNJ2zGSCW/hxldx40Jj3OrOt7HALBT8kyZf/nNOTs/vR5wp7Tjf1tLyyuraemYju7m1vbOb29uvKRFLQqtEcCEbPijKWUirmmlOG5GkEPic1v3h1aRev6dSMRHe6lFE2wH0Q9ZjBLSxOjm7pSHuJJ43xi0R0D6kDFEkxQMuFVxcxHBylxQdu3Q27uTyju1MhRfBTSGPUlU6ua9WV5A4oKEmHJRquk6k2wlIzQin42wrVjQCMoQ+bRoMIaCqnUzvGuNj43RxT0jzQo2n7u+JBAKlRoFvOgPQAzVfm5j/1Zqx7l20ExZGsaYhmS3qxRxrgSch4S6TlGg+MgBEMvNXTAYggWgTZdaE4M6fvAg1z3ZPbe+mlC9fpnFk0CE6QgXkonNURteogqqIoEf0jF7Rm/VkvVjv1sesdclKZw7QH1mfP8odnSA=</latexit>
Example for a black hole with spin a = 0.8M
Frequency and damping time
determined by — and thus encode —
the mass and spin of the final black hole.
Example of ringdown
First event shows some of the best evidence of this structure
Last few cycles of GW150914 are consistent with
this structure, for a final black hole with a/M = 0.7
More events will more fully explore parameter space
Example: Recent work from my group (arXiv:1901.05900;
S.A. Hughes, A. Apte, G. Khanna, H. Lim),
showing how the spectrum of mode
excitation depending on the geometry of the final plunge.
Can also include source … lets us study binaries with one
member much larger than other.