Lecture notes: https://wwwmpa.mpa-garching.mpg.de/~komatsu/lectures--reviews.html Day 3: Sourced Contribution Eiichiro Komatsu [Max Planck Institute for Astrophysics] University of Amsterdam March 19, 2020
We continue to use D ij for the gravitation wave : Newton’s gravitational potential : Spatial scalar curvature perturbation : Tensor metric perturbation [=gravitational waves]
<latexit sha1_base64="tdksEbVCSBH2p0QvRH1PO8w3WZo=">ACHicbVDLSsNAFJ34rPUVdenCwSK4sSRSHxuhVKEuK/QFTQyT6bQdO3kwMxFLyNKNv+LGhSJu/QR3/o3TNAtPXDhzDn3MvceN2RUSMP41ubmFxaXlnMr+dW19Y1NfWu7KYKIY9LAQt420WCMOqThqSkXbICfJcRlru8HLst+4JFzTw63IUEtDfZ/2KEZSY6+Z1WCB3jlxPQuTgyT62Qwmo9fd7G1Vbi6AWjaKSAs8TMSAFkqDn6l9UNcOQRX2KGhOiYRijtGHFJMSNJ3oECREeoj7pKOojwg7Tg9J4IFSurAXcFW+hKn6eyJGnhAjz1WdHpIDMe2Nxf+8TiR753ZM/TCSxMeTj3oRgzKA41Rgl3KCJRspgjCnaleIB4gjLFV2eRWCOX3yLGkeF81S8eSmVChXsjhyYBfsg0NgjNQBtegBhoAg0fwDF7Bm/akvWjv2sekdU7LZnbAH2ifPzjymNs=</latexit> <latexit sha1_base64="UlizORGXUK049JcvMBMihvkQGqQ=">AB9HicbVBNT8JAEJ3iF+IX6tHLRmLibQGo0eiBz1iYoEKtkuW1jZbuvuloQ0/R1ePGiMV3+MN/+NC/Sg4EsmeXlvJjPz/JgzpW372yqsrK6tbxQ3S1vbO7t75f2DpoSahLIh7Jto8V5UxQVzPNaTuWFIc+py1/dD31W2MqFYvEvZ7E1AvxQLCAEayN5HVj1kvZY/aQ3rSyXrliV+0Z0DJxclKBHI1e+avbj0gSUqEJx0p1HDvWXoqlZoTrNRNFI0xGeEB7RgqcEiVl86OztCJUfoiKQpodFM/T2R4lCpSeibzhDroVr0puJ/XifRwaWXMhEnmgoyXxQkHOkITRNAfSYp0XxiCaSmVsRGWKJiTY5lUwIzuLy6R5VnVq1fO7WqV+lcdRhCM4hlNw4ALqcAsNcIHAEzDK7xZY+vFerc+5q0FK585hD+wPn8AIqSVw=</latexit> <latexit sha1_base64="r4VQiey+7DIgHOj3TVxp2xcN35w=">AB+3icbZDLSsNAFIZP6q3W6xLN8EiuCpJqehGKLpxWaE3aGOYTKft2MkzEzEvIqblwo4tYXcefbOE2z0NYfBj7+cw7nzO9HjEpl29GYW19Y3OruF3a2d3bPzAPyx0ZxgKTNg5ZKHo+koRTtqKkZ6kSAo8Bnp+tObeb37SISkIW+pWUTcAI05HVGMlLY8s9zyEvqQXqH72iCiGXtmxa7amaxVcHKoQK6mZ34NhiGOA8IVZkjKvmNHyk2QUBQzkpYGsSQRwlM0Jn2NHAVEukl2e2qdamdojUKhH1dW5v6eSFAg5SzwdWeA1EQu1+bmf7V+rEaXbkJ5FCvC8WLRKGaWCq15ENaQCoIVm2lAWFB9q4UnSCsdFwlHYKz/OV6NSqTr16flevNK7zOIpwDCdwBg5cQANuoQltwPAEz/AKb0ZqvBjvxseitWDkM0fwR8bnDwcLlHI=</latexit> Are GWs from vacuum fluctuation in spacetime, or from sources? π GW ⇤ D ij = − 16 π GT GW ij ij T ij = a 2 π ij • Homogeneous solution : “GWs from the vacuum fluctuation” • We covered this on Day 1 • Inhomogeneous solution : “GWs from sources” • Topic of today’s lecture
<latexit sha1_base64="Be+fbWeIBIsEyP5AuCfKgCuiXI=">ACDHicbVC9TsMwGHT4LeWvwMhiUSGYqgSKYKlUwcJYEP2RmjRyXKd1ayeR7SCqKA/AwquwMIAQKw/Axtvgthmg5ZMsn+7uk3nRYxKZrfxsLi0vLKam4tv76xubVd2NltyDAWmNRxyELR8pAkjAakrqhipBUJgrjHSNMbXo315j0RkobBnRpFxOGoF1CfYqQ05RaKDx1qxDqK0mPKraMuZsMKlbaOb3tUHeghYF2mSVzMnAeWBkogmxqbuHL7oY45iRQmCEp25YZKSdBQlHMSJq3Y0kihIeoR9oaBogT6STMCk81EwX+qHQJ1Bwv7eSBCXcsQ97eRI9eWsNib/09qx8i+chAZRrEiApw/5MYM6/bgZ2KWCYMVGiAsqP4rxH0kEFa6v7wuwZqNPA8aJyWrXDq7KRerl1kdObAPDsAxsMA5qIJrUAN1gMEjeAav4M14Ml6Md+Njal0wsp098GeMzx8Pp5r9</latexit> Which sources? • Scalar, vector, tensor decomposition • When the unperturbed space is homogeneous and isotropic, we can classify perturbations based on how they transform under spatial rotation: 3 X x i → x i 0 = R i j x j • Spin 0: Scalar • Spin 1: Vector j =1 • Spin 2: Tensor
<latexit sha1_base64="jIvgA0heKXkLiQyZiPsrA4UGPA=">ACFXicbVDLSsNAFJ3UV62vqEs3g0VsQUoiFd0IRTcuK9gHNKFMJpN26GQSZiZiCf0JN/6KGxeKuBXc+TdO24jaemDgcM653LnHixmVyrI+jdzC4tLySn61sLa+sblbu80ZQITBo4YpFoe0gSRjlpKoYaceCoNBjpOUNLsd+65YISN+o4YxcUPU4zSgGCktdc2joJQ6XgDvRmVHRdBRlPkDUbf6mH5/CfRNYtWxZoAzhM7I0WQod41Pxw/wklIuMIMSdmxrVi5KRKYkZGBSeRJEZ4gHqkoylHIZFuOrlqBA+04sMgEvpxBSfq74kUhVIOQ08nQ6T6ctYbi/95nUQFZ25KeZwowvF0UZAwqO8fVwR9KghWbKgJwoLqv0LcRwJhpYs6BLs2ZPnSfO4YlcrJ9fVYu0iqyMP9sA+KAEbnIauAJ10AY3INH8AxejAfjyXg13qbRnJHN7I/MN6/AJflndU=</latexit> <latexit sha1_base64="Be+fbWeIBIsEyP5AuCfKgCuiXI=">ACDHicbVC9TsMwGHT4LeWvwMhiUSGYqgSKYKlUwcJYEP2RmjRyXKd1ayeR7SCqKA/AwquwMIAQKw/Axtvgthmg5ZMsn+7uk3nRYxKZrfxsLi0vLKam4tv76xubVd2NltyDAWmNRxyELR8pAkjAakrqhipBUJgrjHSNMbXo315j0RkobBnRpFxOGoF1CfYqQ05RaKDx1qxDqK0mPKraMuZsMKlbaOb3tUHeghYF2mSVzMnAeWBkogmxqbuHL7oY45iRQmCEp25YZKSdBQlHMSJq3Y0kihIeoR9oaBogT6STMCk81EwX+qHQJ1Bwv7eSBCXcsQ97eRI9eWsNib/09qx8i+chAZRrEiApw/5MYM6/bgZ2KWCYMVGiAsqP4rxH0kEFa6v7wuwZqNPA8aJyWrXDq7KRerl1kdObAPDsAxsMA5qIJrUAN1gMEjeAav4M14Ml6Md+Njal0wsp098GeMzx8Pp5r9</latexit> Which sources? • Scalar, vector, tensor decomposition • When the unperturbed space is homogeneous and isotropic, we can classify perturbations based on how they transform under spatial rotation: 3 X x i → x i 0 = R i j x j • Spin 0: Scalar f ( x ) → ˜ j =1 f ( x 0 ) = f ( x ) • Spin 1: Vector • Spin 2: Tensor
<latexit sha1_base64="Be+fbWeIBIsEyP5AuCfKgCuiXI=">ACDHicbVC9TsMwGHT4LeWvwMhiUSGYqgSKYKlUwcJYEP2RmjRyXKd1ayeR7SCqKA/AwquwMIAQKw/Axtvgthmg5ZMsn+7uk3nRYxKZrfxsLi0vLKam4tv76xubVd2NltyDAWmNRxyELR8pAkjAakrqhipBUJgrjHSNMbXo315j0RkobBnRpFxOGoF1CfYqQ05RaKDx1qxDqK0mPKraMuZsMKlbaOb3tUHeghYF2mSVzMnAeWBkogmxqbuHL7oY45iRQmCEp25YZKSdBQlHMSJq3Y0kihIeoR9oaBogT6STMCk81EwX+qHQJ1Bwv7eSBCXcsQ97eRI9eWsNib/09qx8i+chAZRrEiApw/5MYM6/bgZ2KWCYMVGiAsqP4rxH0kEFa6v7wuwZqNPA8aJyWrXDq7KRerl1kdObAPDsAxsMA5qIJrUAN1gMEjeAav4M14Ml6Md+Njal0wsp098GeMzx8Pp5r9</latexit> Which sources? • Scalar, vector, tensor decomposition • When the unperturbed space is homogeneous and isotropic, we can classify perturbations based on how they transform under spatial rotation: 3 X x i → x i 0 = R i j x j • Spin 0: Scalar j =1 x 3 • Spin 1: Vector • Spin 2: Tensor x 2 x 1 (v 1 ,v 2 ,0)
<latexit sha1_base64="MWbhNbIYeGOpABC7zLoY3fEnDEY=">AB7nicbVBNSwMxEJ3Ur1q/qh69BIvgqexKRY9FLx4r2A9ol5JNs21oNhuSbKEs/RFePCji1d/jzX9j2u5BWx8MPN6bYWZeqAQ31vO+UWFjc2t7p7hb2ts/ODwqH5+0TJqypo0EYnuhMQwSVrWm4F6yjNSBwK1g7H93O/PWHa8EQ+2aliQUyGkecEukdm9CtBrxfrniVb0F8Drxc1KBHI1+as3SGgaM2mpIMZ0fU/ZICPacirYrNRLDVOEjsmQdR2VJGYmyBbnzvCFUwY4SrQrafFC/T2RkdiYaRy6zpjYkVn15uJ/Xje10W2QcalSyRdLopSgW2C57/jAdeMWjF1hFDN3a2Yjogm1LqESi4Ef/XldK6qvq16vVjrVK/y+MowhmcwyX4cAN1eIAGNIHCGJ7hFd6Qi/oHX0sWwsonzmFP0CfP3ij64=</latexit> <latexit sha1_base64="yQvLA2K6U1oi6xWUxgyVpcqjyNM=">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</latexit> <latexit sha1_base64="Be+fbWeIBIsEyP5AuCfKgCuiXI=">ACDHicbVC9TsMwGHT4LeWvwMhiUSGYqgSKYKlUwcJYEP2RmjRyXKd1ayeR7SCqKA/AwquwMIAQKw/Axtvgthmg5ZMsn+7uk3nRYxKZrfxsLi0vLKam4tv76xubVd2NltyDAWmNRxyELR8pAkjAakrqhipBUJgrjHSNMbXo315j0RkobBnRpFxOGoF1CfYqQ05RaKDx1qxDqK0mPKraMuZsMKlbaOb3tUHeghYF2mSVzMnAeWBkogmxqbuHL7oY45iRQmCEp25YZKSdBQlHMSJq3Y0kihIeoR9oaBogT6STMCk81EwX+qHQJ1Bwv7eSBCXcsQ97eRI9eWsNib/09qx8i+chAZRrEiApw/5MYM6/bgZ2KWCYMVGiAsqP4rxH0kEFa6v7wuwZqNPA8aJyWrXDq7KRerl1kdObAPDsAxsMA5qIJrUAN1gMEjeAav4M14Ml6Md+Njal0wsp098GeMzx8Pp5r9</latexit> Which sources? • Scalar, vector, tensor decomposition • When the unperturbed space is homogeneous and isotropic, we can classify perturbations based on how they transform under spatial rotation: 3 X x i → x i 0 = R i j x j • Spin 0: Scalar j =1 x 3’ • Spin 1: Vector cos ϕ sin ϕ 0 v ( x 0 ) = v ( x ) → ˜ − sin ϕ cos ϕ 0 v ( x ) x 2’ • Spin 2: Tensor 0 0 1 ϕ x 1’ (~v 1 ,~v 2 ,0)
<latexit sha1_base64="MWbhNbIYeGOpABC7zLoY3fEnDEY=">AB7nicbVBNSwMxEJ3Ur1q/qh69BIvgqexKRY9FLx4r2A9ol5JNs21oNhuSbKEs/RFePCji1d/jzX9j2u5BWx8MPN6bYWZeqAQ31vO+UWFjc2t7p7hb2ts/ODwqH5+0TJqypo0EYnuhMQwSVrWm4F6yjNSBwK1g7H93O/PWHa8EQ+2aliQUyGkecEukdm9CtBrxfrniVb0F8Drxc1KBHI1+as3SGgaM2mpIMZ0fU/ZICPacirYrNRLDVOEjsmQdR2VJGYmyBbnzvCFUwY4SrQrafFC/T2RkdiYaRy6zpjYkVn15uJ/Xje10W2QcalSyRdLopSgW2C57/jAdeMWjF1hFDN3a2Yjogm1LqESi4Ef/XldK6qvq16vVjrVK/y+MowhmcwyX4cAN1eIAGNIHCGJ7hFd6Qi/oHX0sWwsonzmFP0CfP3ij64=</latexit> <latexit sha1_base64="mlmZ9PkmPKmdx03AqR/AwRx+hdA=">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</latexit> <latexit sha1_base64="Be+fbWeIBIsEyP5AuCfKgCuiXI=">ACDHicbVC9TsMwGHT4LeWvwMhiUSGYqgSKYKlUwcJYEP2RmjRyXKd1ayeR7SCqKA/AwquwMIAQKw/Axtvgthmg5ZMsn+7uk3nRYxKZrfxsLi0vLKam4tv76xubVd2NltyDAWmNRxyELR8pAkjAakrqhipBUJgrjHSNMbXo315j0RkobBnRpFxOGoF1CfYqQ05RaKDx1qxDqK0mPKraMuZsMKlbaOb3tUHeghYF2mSVzMnAeWBkogmxqbuHL7oY45iRQmCEp25YZKSdBQlHMSJq3Y0kihIeoR9oaBogT6STMCk81EwX+qHQJ1Bwv7eSBCXcsQ97eRI9eWsNib/09qx8i+chAZRrEiApw/5MYM6/bgZ2KWCYMVGiAsqP4rxH0kEFa6v7wuwZqNPA8aJyWrXDq7KRerl1kdObAPDsAxsMA5qIJrUAN1gMEjeAav4M14Ml6Md+Njal0wsp098GeMzx8Pp5r9</latexit> Which sources? • Scalar, vector, tensor decomposition • When the unperturbed space is homogeneous and isotropic, we can classify perturbations based on how they transform under spatial rotation: 3 X x i → x i 0 = R i j x j • Spin 0: Scalar j =1 x 3’ • Spin 1: Vector spin 1 ( v 1 ± iv 2 )( x ) → (˜ v 1 ± i ˜ v 2 )( x 0 ) = e ⌥ i ϕ ( v 1 ± iv 2 )( x ) x 2’ • Spin 2: Tensor ϕ x 1’ (~v 1 ,~v 2 ,0)
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