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

◆

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@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=">AAACOnicbVDJSgNBEO2JW4xb1KOXxiBED2EmCnoRgl5yTMAskAlDT6cnadKz0F0jhGG+y4tf4c2DFw+KePUD7CxgFh80vHqviup6biS4AtN8NTJr6xubW9nt3M7u3v5B/vCoqcJYUtagoQhl2yWKCR6wBnAQrB1JRnxXsJY7vB/7rUcmFQ+DBxhFrOuTfsA9TgloycnXbU8SmtgRkcCJwFXHTP8qSPEtXuyYc6VzkdqCeVDUDNdtyfsDOHfyBbNkToBXiTUjBTRDzcm/2L2Qxj4LgAqiVMcyI+gm4yVUsDRnx4pFhA5Jn3U0DYjPVDeZnJ7iM630sBdK/QLAE3V+IiG+UiPf1Z0+gYFa9sbif14nBu+mm/AgioEFdLrIi3UkIR7niHtcMgpipAmhkuu/YjogOinQaed0CNbyyaukWS5Zl6Vy/apQuZvFkUUn6BQVkYWuUQVVUQ01EEVP6A19oE/j2Xg3vozvaWvGmM0cowUYP78gWq4u</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=">AAACLXicbVBNSxxBFOzR+LV+rebopckSUNHdGRUUJCCaQ44Krgo76/Cm9+3a2D0zdr8RlmH/kJf8lRDIQZFc8zfSs84hagoaiqp6vH4VZ0pa8v1Hb2Lyw9T0zOxcbX5hcWm5vrJ6YdPcCGyLVKXmKgaLSibYJkkKrzKDoGOFl/HtSelf3qOxMk3OaZhhV8MgkX0pgJwU1b+e8dBKzfG62KZWSJCPSi7DVOMAIrNOfJubaHNjFB5uhXd3OfR4meJfeNCqQjKqN/ymPwZ/T4KKNFiF06j+M+ylIteYkFBgbSfwM+oWYEgKhaNamFvMQNzCADuOJqDRdovxtSP+2Sk93k+NewnxsfrvRAHa2qGOXVID3di3Xin+z+vk1D/oFjLJcsJEvCzq54pTysvqeE8aFKSGjoAw0v2VixswIMgVXHMlBG9Pfk8udprBbnPnbK9xdFzVMcvW2Ce2zgK2z47YN3bK2kywB/aDPbIn77v3y3v2fr9EJ7xq5iN7Be/PX3ufpnA=</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=">AAAC7HicdVJNj9MwEHXC1xK+unDkYlGBuqJUSUBajivgwHGR6O5KdVpNXaex1nGy9mSlKspv4MIBhLjyg7jxb3CbSLC73ZFsPb+ZeTMee14qaTEM/3j+jZu3bt/ZuRvcu//g4aPe7uMjW1SGizEvVGFO5mCFklqMUaISJ6URkM+VOJ6fvl/7j8+FsbLQn3FViiSHpZap5ICOmu16vpoyUGUG9AVbFCjOKEsN8DpqavZBKISGKZHiZGCmMX1JYRrvDVvHMBwCM3KZYRIwFuhrdeJ/uYwX1u2YCYS9Lcqvtkrn17dozwzWcTMwTkFSWOtfVJfArNQt52SjoWTc8vbclZj1+uEo3Bi9CqIO9Elnh7Peb9cHr3KhkSuwdhKFJSY1GJRciSZglRUl8FNYiomDGnJhk3rzWA197pgFTQvjlka6Yf/PqCG3dpXPXWQOmNnLvjW5zTepMH2b1FKXFQrN20JppSgWdP3ydCGN4KhWDgA30vVKeQZukOj+R+CGEF2+8lVwFI+i16P405v+wbtuHDvkKXlGBiQi++SAfCSHZEy4J70v3jfvu6/9r/4P/2cb6ntdzhNywfxffwERG+ZG</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=">AAACNnicbVDBSiNBFOxRV92460Y9emkMC142zOiCXgTRixchwkaFTDK86bwkjd09Q/cbIQz5Ki9+hzcvHnYRr36CnUkOu7oFDUVVPV6/SnMlHYXhY7CwuPRpeWX1c23ty9f1b/WNzUuXFVZgW2Qqs9cpOFTSYJskKbzOLYJOFV6lN6dT/+oWrZOZ+UXjHLsahkYOpADyUlI/j1tOJvv8iP84TcoYVD6COEWCeAhaQ9xHRTDhqjezuOlVLo9TsFz3ZqmpWgWTeiNshhX4RxLNSYPN0UrqD3E/E4VGQ0KBc50ozKlbgiUpFE5qceEwB3EDQ+x4akCj65bV2RP+3St9Psisf4Z4pf49UYJ2bqxTn9RAI/fem4r/8zoFDQ67pTR5QWjEbNGgUJwyPu2Q96VFQWrsCQgr/V+5GIEFQb7pmi8hen/yR3K514z2m3sXPxvHJ/M6Vtk222G7LGIH7JidsRZrM8Hu2CP7zf4E98FT8By8zKILwXxmi/2D4PUNN2Krqw==</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=">AAACbnicbVHBahsxENVumyZx08ZpoYeGUlETcA4xu4mhvRRCc8nRaeskYDmLVp61RaTVIs0WzLLH/mBv/YZe+gmR7S0kTh8IHu/NjDRPaaGkwyj6HYRPnm4829zabj3fefFyt7336tKZ0goYCqOMvU65AyVzGKJEBdeFBa5TBVfp7dnCv/oB1kmTf8d5AWPNp7nMpODopaT9kw2cTPr0M2WZ5aKK66pr6RGVnAnjGM4A+eFNv2YyRzphRsOUU+ZKnVQMlKK6/vqPrcy6aw+/rUvNHLippGbFTNbU0yPZjMM6aXeiXrQEfUzihnRIg0HS/sUmRpQachSKOzeKowLHFbcohYK6xUoHBRe3fAojT3OuwY2rZVw1PfDKhGbG+uO3Wqr3OyqunZvr1FdqjjO37i3E/3mjErNP40rmRYmQi9VFWakoGrrInk6kBYFq7gkXVvq3UjHjPnb0P9TyIcTrKz8ml8e9+KR3fNHvnH5p4tgi++QD6ZKYfCSn5JwMyJAI8ifYC94G+8Hf8E34Lny/Kg2Dpuc1eYCwewc1ibud</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

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⌧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.