Finite boundary observables and white holes

Transcription

1 F boudry obsrvbls d wh hols Crlo Rovll F boudry obsrvbls [cr: Do] Cocr clculo obsrvbl: blck wh hol ullg d Fs Rdo Burss

2 Fcs robl uu grvy Mrs 31s Brbr

3 Fcs robl uu grvy Mrs 31s Brbr

4 V 3 3 v r r 0 How bg blck hol? Mros Chrdoulou Crlo Rovll PhysRv D91 ( r v Ar 3 : A d voro: V 5 V 7 rl volu blck hol lrg v wh s Ar-Eroy sll: ur rl dgrs rdo us lrg ordr o vol locl s ow do vldy hologrhy?

5 A uu ory grvy co locl : 1 Locl obsrvbls r o gug vr grvy: d vrc c- dcrss or slr (cr: boh loos d srgs [cr: Gbrl] Locl : dgrs rdo locld wh rsc clsscl s uu grvy: dgrs rdo locld wh rsc o or s clsscl grl rlvy

6 Grl lss ro o rurbv G rl obsrvbls ( locl obsrvbls d gug vrc uu ory gvs robbls oucos (locl rco uu sys wh or sys Grl rlvy dscrs (locl rco sc rgos wh o or

7 Grl lss ro o rurbv G rl obsrvbls ( locl obsrvbls d gug vrc uu ory gvs robbls oucos (locl rco uu sys wh or sys Grl rlvy dscrs (locl rco sc rgos wh o or Co wo scs

8 Grl lss ro o rurbv G rl obsrvbls ( locl obsrvbls d gug vrc uu ory gvs robbls oucos (locl rco uu sys wh or sys Grl rlvy dscrs (locl rco sc rgos wh o or Co wo scs uu sys c rgo Boudry Hl uco: (

9 Grl lss ro o rurbv G rl obsrvbls ( locl obsrvbls d gug vrc uu ory gvs robbls oucos (locl rco uu sys wh or sys Grl rlvy dscrs (locl rco sc rgos wh o or Co wo scs uu sys c rgo Boudry Hl uco: ( Boudry orl s: scd 3d boudrs sc rgos r luds: scd 4d rgos [cr: Bc]

10 0 0 Hl uco ( d L( 0 0 ( 0 0 ( Progr W ( 0 0 h H( 0 0 (0 0 D[] R dl ( ( 0 0 yss volvg obsrvbl ys volvg rr ( ( ( L[ ] L 0 [ ṫ] ṫ ṫ Hl uco Progr ( L( 0 0( 0 0( ṫ 0 0( 0 0( ( 0 0 W ( 0 0 h P 0 0 (0 0 D[ ] R dl( ṫ ( ( 0 0 ( ( 0 0

11 ' Boudry ucol W [' ] ' D [ ] Σ 7

12 ' Boudry ucol W [' ] ' D [ ] Σ Bu grlly covr ory: W [' ] W ['] c rgo Dc d surs Prcl dcr I grl rlvc ory dc d surs r ld surs lk or os: y r drd boudry d robl [cr: J] 7

13 r luds h H( 0 0 R vr h P 0 hw 0 uu grvy hw boudry

14 boudry drs hr d ow rco whos corrlos r rdcd uu ory hyscl rlv o os o boudry drd (uu s gory o boudry sl uu grvy Crlo Rovll CUP 004 Iroduco Cocl Loo uu Grvy Frcsc Vdo Crlo Rovll CUP 014

15 L s k h cocr

16 Covr loo uu grvy Full do 0 Kcs Boudry sc Orrs [cr: v]: H L [U( L /U( ] 3 (h l H l 1 H h l l s work (ods lks Dycs Bulk r luds Vrx lud W C (h l C A(h v Y dh v U( Y dg 0 L(C j (h Y v A(h v W l C1 W C (j + 1 D j (h v D (j+1 j jj (g g 1 0 h Y v h v 8 G 1 v Wh cologcl cos: C (vrcs dgs cs L(C Chr-os ory o-plr Grh Orr d 4D Loo uu Grvy wh Cologcl Cos Hl M Hggrd Mux H Wojcch Kńsk Aldo Rllo rv:

17 P M rsuls: ( Clsscl l: GR ( c dcrss ( UV d IR luds Exc uu grvy r luds W (h l 0 Grl rlvy Hl uco ( Couous l C1 LG r luds W C (h l 0 1 Rgg Hl uco (l b Clsscl l Rg vldy x:

18 (8 dcrz dcrzd h or dll Cosdr syw w hv hyscl sros kg o k couu sd 1 II DICREIAIO III II vry slr grl rlvy lrg ur sll ss sz / bu hd lrg ur sll ss sz / II DICREIAIO hyscl s s or dlss vrbls sys ky obsrvo s s crcl o c l uvrsly d o suc: w us l sd s (+1 ( (3 ICREIAIO hyscl sros kg dcrzd sysvr udr s rrrzo couu ory wll rcovrd wh 1 d couu ory wll rcovrd wh dcrzo 1 d crcl d oc h s r o dd usg uous uos s oo (8 whch gvs dly hroc oscllr uo wsh wy c dls crcl vlu 0 sc d us go zro r hroc oscllr wh ss d guwhy h works? D-vrc (ro B Drch Cosdr s s o c l uvrsly Cosdr uslss vrbls (D ro vrc kg sz uo rvl crcl 0 kg sz wh rvl xd wsh wy rs c srcly whch would h cox ( ollow h hvor rs ldcoord ory Iscod xd url d0 /d ; whl or d gu l Mor sy ww oscllr cou Cosdr s hroc ss d cy co ( As volv ss rhs ow dlss wy c dls dcrzo d ( volv L ushgs h d sys Dcrz sys o ss dd wh Dcrz sys o ss dcrz r wh cus oscllr o wh r d guh s hvor sc wsh olr rucy bu rr url us uos rs co hk h h r uvrsl Bu drogr co couo ory rd c o (or r Euh s hvor 1 rs ld I rd h url dscrd sl ss sz gr wh sys d sys gr 1 s dscrd co s lycl hk h gv dycs sl ( grl rlvy s or r dcrzg rrrzo-vr sys d hgs c r 0 dcrzd (9 ( Cosd + uvrsl sudd d url (1 c urclly gv hk h h r Bu d ( d vrbls ( d co s vrbls ( d co c dcrzd s H( r wh dcr us r rovdd Plck lgh h s d yss W ( ; h d d volvg obsrvbl co sys- (1 o r wh dcrzg rrrzo-vr d h grl v+1 v (+1 ollows ollows s ylds dlss co d ( d (1 whch o dd uo: s sly d rvl [(] +1 xd rs d dvd D[(] (5 ( [(] ( whr dcr cosrvo rgy h ro sys- vlocy ollows (8 III PARAMERIAIO D[(] sll (4 (+1 1 ( Choos xd rvl rs d dvd ur Dcrz ssd sz / ( ( whr do whr do +1 (+1 gug vrc udr hs dd lrg rbrry III P rvl rs d dvd lrg ur sll ss sz / I ory wll rcovrd wh 1 d I d d III PARAMERIAIO h gv rrrzo s dd vrbl I h v sll ss sz hv / +1 Cosdr dd wh h couu sd 1 couu ory wllsys rcovrd wo vrbls 1 d ( l g sz l w rvl xd hr l u hr +1 ( (1 A sdrd lc rocdur d rscld oc hg or: A sdrd lc rocdur d rscld vry slr grl rlvy whch ully ls us ss 1 sys ( volvg ( voluo rr d gov- Cosdr s 0 d kg sz rvl xd orcovrd suc: w ld sys owh sd wh sco I c sco I c dlss vrbls d h xrs Idd bov rd dlss vrbls d W ( ; l d (6 vr udr rrrzo s dd Cosdr sys dd wo vrbls ( lu 1 0 sys volvg rd sys co d wh xdd sz( s Dcrz o ss sc rvl d dscrd us goh xd zro oc rucy brd or0 hs h dlss co cos h 1 dlss co srd coord 1 (D vrc d ( volvg dscrd d govo w sz (+1 Wh cos gr vrbls voluo sys ( dss sy co c s Mor rcly wwh dcrzd cou rr co lzo dlss vrbls uos ow ( +1 As L us dcrz hco sys or x rvl rd co dcrzd (+ w w o? vro co vrbls ( d c s 1 orh sys dscrd 1 s d (7 cou r lud s whr 0 d subl +1 (3 ( ( d sly ( (3 +1 sl ss sz d +1 ( d co c dcrzd s ollows sur or cr or orlzo whch gvs whch gvs ys volvg ( d ( Cosdr dcrzd +1 H( (7 rr h /d c h (lwhr od o oly kg I words dcrzd co ully do dc drvv wh rsc d (1 ; d /d +1 r co E+1 E dlss ( 1 bu l sdg whr ll us o ow whr ll us o rhs r ow dlss (1 rhs crcl ( s vlu ss h lry obsrvo whr do dc [( ( ] I d s h h hysclly ully uv +1 W ( ; D[( ]D[( ] [(] h co (or r lycl couo Eus crh co (or r s lycl couo udy Eu-I (0 +1 l dg ( D[(] ur (5 hroc whr do dc drvv wh rsc rcl L us cosucs dcrzo r d s l oscllr dcussd rvous ( h rgy ow cosrv +1 Dcrz rocdur rd lc d rscld (0 ( cld c sudd urclly gv uv (10 cld c sudd gv I d s h h hysclly ully lghs dcrzd sys cosuc c l h hro sco I c uo oo or +1 urclly cl vlu corrlo I ossbl cus dd (13 A sdrd lc rocdur d rscld lss vrbls d roxo h grl r roxo h grl l hroc oscllr dcussd rvous dvrg ur lc ss h y ory o gv doubl sco I c l gv whch o h gv ocdur d rscld dlss co cos dlss vrbls d d or whch whch o o uo oo sco I c (8 dycs ordr or oc hg or: uy dros ro s d h dlss co cos [(] ( cosrvo ( [(] cosrvo o D[(] d (4 D[(] d (4 ( ( I or words coous h xrs Idd bov rds sco ;( cos l d (6 ( (3 W ( (14 +1 ; l d d d hs hs dd dd whch 11 0 hroc (8 dgr rdo whch u gvs dly oscllr uo ( ( ( rrrz rrrz (+1l /d couu uo w o odk k couu l w hv hv sd sd 1 1whchdycs o-rrzd br ; whl or gvs d ( (3 ( (11 whr d d vry sl +1 vry s 0 h o suc: 0w us us o r bu l sd s d rhs ow d dlss whch subl bu h o suc: w us l sd sd /d gvs dly hroc oscllr uo Bu dcr rrzd ; dy wh +1 r o or l k h 1 vr udr vr ud (orcr r s lycl couod Euwhr ll ouo rhs r ow crcl us dsc ur 0 d us go zro o or sur vlu 0 sc ddlss us go ddd zro rrz ss rdo crcl ;vlu whl or yss volvg Wh : /d couu l od kg y + 0 (9 coord lrg vrg ss r ucoord vr vrd c l sudd urclly gv rhs r ow dlss h co (or r s lycl couo Eu co l Mor rcly sy w w cou h od o oly kg I words dcrzd ully dd ro l Mor rcly sy w w cou d coulg cos crcl vlu xg h rgy cosrv clly sll cld dc L L us us dcr dcr h grl o s lycl couo Eu sudd urclly gv - rogr u l sdg s crcl vlu ss- rogr h lry o hrd s obsrvo + 0 (9 Blow I sudy whr clsscl d u sl whch o dd uo: sly sl r covr Grl crcouu l kg jus ur ss y sudd urclly gv syss: roxo od us h grl rcl L sudy cosucs h c g ur dcrzo r dcr ory: y u such dcrzo ro (8 sys (dycs ( cosrvo rgy h ollows whch d o ( H( hd[(] grl H( d [(] d (4 gv dcrzd sys corrlo lghs cosuc h l W ( h djus cosrvo rgy W ( h ;; couu rrzd ory rd[(] wlldd dd d [(] sly ( hs lrg udr rbrry co o dd uo: vrc lc ss h ssbl d (4 gug ur ywhch r ory o gv doubl l 1 cosrvo co gro dcr ( ( [(] rrrzo s dd vrbl I h hs dd l rgy h ollows ro (8 sys[(] d l [(] h couu w(hv (4 sd cosrvo 1

19 (8 dcrz dcrzd h or dll Cosdr syw w hv hyscl sros kg o k couu sd 1 II DICREIAIO III II vry slr grl rlvy lrg ur sll ss sz / bu hd lrg ur sll ss sz / II DICREIAIO hyscl s s or dlss vrbls sys ky obsrvo s s crcl o c l uvrsly d o suc: w us l sd s (+1 ( (3 ICREIAIO hyscl sros kg dcrzd sysvr udr s rrrzo couu ory wll rcovrd wh 1 d couu ory wll rcovrd wh dcrzo 1 d crcl d oc h s r o dd usg uous uos s oo (8 whch gvs dly hroc oscllr uo wsh wy c dls crcl vlu 0 sc d us go zro r hroc oscllr wh ss d guwhy h works? D-vrc (ro B Drch Cosdr s s o c l uvrsly Cosdr uslss vrbls (D ro vrc kg sz uo rvl crcl 0 kg sz wh rvl xd wsh wy rs c srcly whch would h cox ( ollow h hvor rs ldcoord ory Iscod xd url d0 /d ; whl or d gu l Mor sy ww oscllr cou Cosdr s hroc ss d cy co ( As volv ss rhs ow dlss wy c dls dcrzo d ( volv L ushgs h d sys Dcrz sys o ss dd wh Dcrz sys o ss dcrz r wh cus oscllr o wh r d guh s hvor sc wsh olr rucy bu rr url us uos rs co hk h h r uvrsl Bu drogr co couo ory rd c o (or r Euh s hvor 1 rs ld I rd h url dscrd sl ss sz gr wh sys d sys gr 1 s dscrd co s lycl hk h gv dycs sl ( grl rlvy s or r dcrzg rrrzo-vr sys d hgs c r 0 dcrzd (9 ( Cosd + uvrsl sudd d url (1 c urclly gv hk h h r Bu d ( d vrbls ( d co s vrbls ( d co c dcrzd s H( r wh dcr us r rovdd Plck lgh h s d yss W ( ; h d d volvg obsrvbl co sys- (1 o r wh dcrzg rrrzo-vr d h grl v+1 v (+1 ollows ollows s ylds dlss co d ( d (1 whch o dd uo: s sly d rvl [(] +1 xd rs d dvd D[(] (5 ( [(] ( whr dcr cosrvo rgy h ro sys- vlocy ollows (8 III PARAMERIAIO D[(] sll (4 (+1 1 ( Choos xd rvl rs d dvd ur Dcrz ssd sz / ( ( whr do whr do +1 (+1 gug vrc udr hs dd lrg rbrry III P rvl rs d dvd lrg ur sll ss sz / I ory wll rcovrd wh 1 d I d d III PARAMERIAIO h gv rrrzo s dd vrbl I h v sll ss sz hv / +1 Cosdr dd wh h couu sd 1 couu ory wllsys rcovrd wo vrbls 1 d ( l g sz l w rvl xd hr l u hr +1 ( (1 A sdrd lc rocdur d rscld oc hg or: A sdrd lc rocdur d rscld vry slr grl rlvy whch ully ls us ss 1 sys ( volvg ( voluo rr d gov- Cosdr s 0 d kg sz rvl xd orcovrd suc: w ld sys owh sd wh sco I c sco I c dlss vrbls d h xrs Idd bov rd dlss vrbls d W ( ; l d (6 vr udr rrrzo s dd Cosdr sys dd wo vrbls ( lu 1 0 sys volvg rd sys co d wh xdd sz( s Dcrz o ss sc rvl d dscrd us goh xd zro oc rucy brd or0 hs h dlss co cos h 1 dlss co srd coord 1 (D vrc d ( volvg dscrd d govo w sz (+1 Wh cos gr vrbls voluo sys ( dss sy co c s Mor rcly wwh dcrzd cou rr co lzo dlss vrbls uos ow ( +1 As L us dcrz hco sys or x rvl rd co dcrzd (+ w w o? vro co vrbls ( d c s 1 orh sys dscrd 1 s d (7 cou r lud s whr 0 d subl +1 (3 ( ( d sly ( (3 +1 sl ss sz d +1 ( d co c dcrzd s ollows sur or cr or orlzo whch gvs whch gvs ys volvg ( d ( Cosdr dcrzd +1 H( (7 rr h /d c h (lwhr od o oly kg I words dcrzd co ully do dc drvv wh rsc d (1 ; d /d +1 r co E+1 E dlss ( 1 bu l sdg whr ll us o ow whr ll us o rhs r ow dlss (1 rhs crcl ( s vlu ss h lry obsrvo whr do dc [( ( ] I d s h h hysclly ully uv +1 W ( ; D[( ]D[( ] [(] h co (or r lycl couo Eus crh co (or r s lycl couo udy Eu-I (0 +1 l dg ( D[(] ur (5 hroc whr do dc drvv wh rsc rcl L us cosucs dcrzo r d s l oscllr dcussd rvous ( h rgy ow cosrv +1 Dcrz rocdur rd lc d rscld (0 ( cld c sudd urclly gv uv (10 cld c sudd gv I d s h h hysclly ully lghs dcrzd sys cosuc c l h hro sco I c uo oo or +1 urclly cl vlu corrlo I ossbl cus dd (13 A sdrd lc rocdur d rscld lss vrbls d roxo h grl r roxo h grl l hroc oscllr dcussd rvous dvrg ur lc ss h y ory o gv doubl sco I c l gv whch o h gv ocdur d rscld dlss co cos dlss vrbls d d or whch whch o o uo oo sco I c (8 dycs ordr or oc hg or: uy dros ro s d h dlss co cos [(] ( cosrvo ( [(] cosrvo o D[(] d (4 D[(] d (4 ( ( I or words coous h xrs Idd bov rds sco ;( cos l d (6 ( (3 W ( (14 +1 ; l d d d hs hs dd dd whch 11 0 hroc (8 dgr rdo whch u gvs dly oscllr uo ( ( ( rrrz rrrz (+1l /d couu uo w o odk k couu l w hv hv sd sd 1 1whchdycs o-rrzd br ; whl or gvs d ( (3 ( (11 whr d d vry sl +1 vry s 0 h o suc: 0w us us o r bu l sd s d rhs ow d dlss whch subl bu h o suc: w us l sd sd /d gvs dly hroc oscllr uo Bu dcr rrzd ; dy wh +1 r o or l k h 1 vr udr vr ud (orcr r s lycl couod Euwhr ll ouo rhs r ow crcl us dsc ur 0 d us go zro o or sur vlu 0 sc ddlss us go ddd zro rrz ss rdo crcl ;vlu whl or yss volvg Wh : /d couu l od kg y + 0 (9 coord lrg vrg ss r ucoord vr vrd c l sudd urclly gv rhs r ow dlss h co (or r s lycl couo Eu co l Mor rcly sy w w cou h od o oly kg I words dcrzd ully dd ro l Mor rcly sy w w cou d coulg cos crcl vlu xg h rgy cosrv clly sll cld dc L L us us dcr dcr h grl o s lycl couo Eu sudd urclly gv - rogr u l sdg s crcl vlu ss- rogr h lry o hrd s obsrvo + 0 (9 Blow I sudy whr clsscl d u sl whch o dd uo: sly sl r covr Grl crcouu l kg jus ur ss y sudd urclly gv syss: roxo od us h grl rcl L sudy cosucs h c g ur dcrzo r dcr ory: y u such dcrzo ro (8 sys (dycs ( cosrvo rgy h ollows whch d o ( H( hd[(] grl H( d [(] d (4 gv dcrzd sys corrlo lghs cosuc h l W ( h djus cosrvo rgy W ( h ;; couu rrzd ory rd[(] wlldd dd d [(] sly ( hs lrg udr rbrry co o dd uo: vrc lc ss h ssbl d (4 gug ur ywhch r ory o gv doubl l 1 cosrvo co gro dcr ( ( [(] rrrzo s dd vrbl I h hs dd l rgy h ollows ro (8 sys[(] d l [(] h couu w(hv (4 sd cosrvo 1

20 (8 dcrz dcrzd h or dll Cosdr syw w hv hyscl sros kg o k couu sd 1 II DICREIAIO III II vry slr grl rlvy lrg ur sll ss sz / bu hd lrg ur sll ss sz / II DICREIAIO hyscl s s or dlss vrbls sys ky obsrvo s s crcl o c l uvrsly d o suc: w us l sd s (+1 ( (3 ICREIAIO hyscl sros kg dcrzd sysvr udr s rrrzo couu ory wll rcovrd wh 1 d couu ory wll rcovrd wh dcrzo 1 d crcl d oc h s r o dd usg uous uos s oo (8 whch gvs dly hroc oscllr uo wsh wy c dls crcl vlu 0 sc d us go zro r hroc oscllr wh ss d guwhy h works? D-vrc (ro B Drch Cosdr s s o c l uvrsly Cosdr uslss vrbls (D ro vrc kg sz uo rvl crcl 0 kg sz wh rvl xd wsh wy rs c srcly whch would h cox ( ollow h hvor rs ldcoord ory Iscod xd url d0 /d ; whl or d gu l Mor sy ww oscllr cou Cosdr s hroc ss d cy co ( As volv ss rhs ow dlss wy c dls dcrzo d ( volv L ushgs h d sys Dcrz sys o ss dd wh Dcrz sys o ss dcrz r wh cus oscllr o wh r d guh s hvor sc wsh olr rucy bu rr url us uos rs co hk h h r uvrsl Bu drogr co couo ory rd c o (or r Euh s hvor 1 rs ld I rd h url dscrd sl ss sz gr wh sys d sys gr 1 s dscrd co s lycl hk h gv dycs sl ( grl rlvy s or r dcrzg rrrzo-vr sys d hgs c r 0 dcrzd (9 ( Cosd + uvrsl sudd d url (1 c urclly gv hk h h r Bu d ( d vrbls ( d co s vrbls ( d co c dcrzd s H( r wh dcr us r rovdd Plck lgh h s d yss W ( ; h d d volvg obsrvbl co sys- (1 o r wh dcrzg rrrzo-vr d h grl v+1 v (+1 ollows ollows s ylds dlss co d ( d (1 whch o dd uo: s sly d rvl [(] +1 xd rs d dvd D[(] (5 ( [(] ( whr dcr cosrvo rgy h ro sys- vlocy ollows (8 III PARAMERIAIO D[(] sll (4 (+1 1 ( Choos xd rvl rs d dvd ur Dcrz ssd sz / ( ( whr do whr do +1 (+1 gug vrc udr hs dd lrg rbrry III P rvl rs d dvd lrg ur sll ss sz / I ory wll rcovrd wh 1 d I d d III PARAMERIAIO h gv rrrzo s dd vrbl I h v sll ss sz hv / +1 Cosdr dd wh h couu sd 1 couu ory wllsys rcovrd wo vrbls 1 d ( l g sz l w rvl xd hr l u hr +1 ( (1 A sdrd lc rocdur d rscld oc hg or: A sdrd lc rocdur d rscld vry slr grl rlvy whch ully ls us ss 1 sys ( volvg ( voluo rr d gov- Cosdr s 0 d kg sz rvl xd orcovrd suc: w ld sys owh sd wh sco I c sco I c dlss vrbls d h xrs Idd bov rd dlss vrbls d W ( ; l d (6 vr udr rrrzo s dd Cosdr sys dd wo vrbls ( lu 1 0 sys volvg rd sys co d wh xdd sz( s Dcrz o ss sc rvl d dscrd us goh xd zro oc rucy brd or0 hs h dlss co cos h 1 dlss co srd coord 1 (D vrc d ( volvg dscrd d govo w sz (+1 Wh cos gr vrbls voluo sys ( dss sy co c s Mor rcly wwh dcrzd cou rr co lzo dlss vrbls uos ow ( +1 As L us dcrz hco sys or x rvl rd co dcrzd (+ w w o? vro co vrbls ( d c s 1 orh sys dscrd 1 s d (7 cou r lud s whr 0 d subl +1 (3 ( ( d sly ( (3 +1 sl ss sz d +1 ( d co c dcrzd s ollows sur or cr or orlzo whch gvs whch gvs ys volvg ( d ( Cosdr dcrzd +1 H( (7 rr h /d c h (lwhr od o oly kg I words dcrzd co ully do dc drvv wh rsc d (1 ; d /d +1 r co E+1 E dlss ( 1 bu l sdg whr ll us o ow whr ll us o rhs r ow dlss (1 rhs crcl ( s vlu ss h lry obsrvo whr do dc [( ( ] I d s h h hysclly ully uv +1 W ( ; D[( ]D[( ] [(] h co (or r lycl couo Eus crh co (or r s lycl couo udy Eu-I (0 +1 l dg ( D[(] ur (5 hroc whr do dc drvv wh rsc rcl L us cosucs dcrzo r d s l oscllr dcussd rvous ( h rgy ow cosrv +1 Dcrz rocdur rd lc d rscld (0 ( cld c sudd urclly gv uv (10 cld c sudd gv I d s h h hysclly ully lghs dcrzd sys cosuc c l h hro sco I c uo oo or +1 urclly cl vlu corrlo I ossbl cus dd (13 A sdrd lc rocdur d rscld lss vrbls d roxo h grl r roxo h grl l hroc oscllr dcussd rvous dvrg ur lc ss h y ory o gv doubl sco I c l gv whch o h gv ocdur d rscld dlss co cos dlss vrbls d d or whch whch o o uo oo sco I c (8 dycs ordr or oc hg or: uy dros ro s d h dlss co cos [(] ( cosrvo ( [(] cosrvo o D[(] d (4 D[(] d (4 ( ( I or words coous h xrs Idd bov rds sco ;( cos l d (6 ( (3 W ( (14 +1 ; l d d d hs hs dd dd whch 11 0 hroc (8 dgr rdo whch u gvs dly oscllr uo ( ( ( rrrz rrrz (+1l /d couu uo w o odk k couu l w hv hv sd sd 1 1whchdycs o-rrzd br ; whl or gvs d ( (3 ( (11 whr d d vry sl +1 vry s 0 h o suc: 0w us us o r bu l sd s d rhs ow d dlss whch subl bu h o suc: w us l sd sd /d gvs dly hroc oscllr uo Bu dcr rrzd ; dy wh +1 r o or l k h 1 vr udr vr ud (orcr r s lycl couod Euwhr ll ouo rhs r ow crcl us dsc ur 0 d us go zro o or sur vlu 0 sc ddlss us go ddd zro rrz ss rdo crcl ;vlu whl or yss volvg Wh : /d couu l od kg y + 0 (9 coord lrg vrg ss r ucoord vr vrd c l sudd urclly gv rhs r ow dlss h co (or r s lycl couo Eu co l Mor rcly sy w w cou h od o oly kg I words dcrzd ully dd ro l Mor rcly sy w w cou d coulg cos crcl vlu xg h rgy cosrv clly sll cld dc L L us us dcr dcr h grl o s lycl couo Eu sudd urclly gv - rogr u l sdg s crcl vlu ss- rogr h lry o hrd s obsrvo + 0 (9 Blow I sudy whr clsscl d u sl whch o dd uo: sly sl r covr Grl crcouu l kg jus ur ss y sudd urclly gv syss: roxo od us h grl rcl L sudy cosucs h c g ur dcrzo r dcr ory: y u such dcrzo ro (8 sys (dycs ( cosrvo rgy h ollows whch d o ( H( hd[(] grl H( d [(] d (4 gv dcrzd sys corrlo lghs cosuc h l W ( h djus cosrvo rgy W ( h ;; couu rrzd ory rd[(] wlldd dd d [(] sly ( hs lrg udr rbrry co o dd uo: vrc lc ss h ssbl d (4 gug ur ywhch r ory o gv doubl l 1 cosrvo co gro dcr ( ( [(] rrrzo s dd vrbl I h hs dd l rgy h ollows ro (8 sys[(] d l [(] h couu w(hv (4 sd cosrvo 1

21 (8 dcrz dcrzd h or dll Cosdr syw w hv hyscl sros kg o k couu sd 1 II DICREIAIO III II vry slr grl rlvy lrg ur sll ss sz / bu hd lrg ur sll ss sz / II DICREIAIO hyscl s s or dlss vrbls sys ky obsrvo s s crcl o c l uvrsly d o suc: w us l sd s (+1 ( (3 ICREIAIO hyscl sros kg dcrzd sysvr udr s rrrzo couu ory wll rcovrd wh 1 d couu ory wll rcovrd wh dcrzo 1 d crcl d oc h s r o dd usg uous uos s oo (8 whch gvs dly hroc oscllr uo wsh wy c dls crcl vlu 0 sc d us go zro r hroc oscllr wh ss d guwhy h works? D-vrc (ro B Drch Cosdr s s o c l uvrsly Cosdr uslss vrbls (D ro vrc kg sz uo rvl crcl 0 kg sz wh rvl xd wsh wy rs c srcly whch would h cox ( ollow h hvor rs ldcoord ory Iscod xd url d0 /d ; whl or d gu l Mor sy ww oscllr cou Cosdr s hroc ss d cy co ( As volv ss rhs ow dlss wy c dls dcrzo d ( volv L ushgs h d sys Dcrz sys o ss dd wh Dcrz sys o ss dcrz r wh cus oscllr o wh r d guh s hvor sc wsh olr rucy bu rr url us uos rs co hk h h r uvrsl Bu drogr co couo ory rd c o (or r Euh s hvor 1 rs ld I rd h url dscrd sl ss sz gr wh sys d sys gr 1 s dscrd co s lycl hk h gv dycs sl ( grl rlvy s or r dcrzg rrrzo-vr sys d hgs c r 0 dcrzd (9 ( Cosd + uvrsl sudd d url (1 c urclly gv hk h h r Bu d ( d vrbls ( d co s vrbls ( d co c dcrzd s H( r wh dcr us r rovdd Plck lgh h s d yss W ( ; h d d volvg obsrvbl co sys- (1 o r wh dcrzg rrrzo-vr d h grl v+1 v (+1 ollows ollows s ylds dlss co d ( d (1 whch o dd uo: s sly d rvl [(] +1 xd rs d dvd D[(] (5 ( [(] ( whr dcr cosrvo rgy h ro sys- vlocy ollows (8 III PARAMERIAIO D[(] sll (4 (+1 1 ( Choos xd rvl rs d dvd ur Dcrz ssd sz / ( ( whr do whr do +1 (+1 gug vrc udr hs dd lrg rbrry III P rvl rs d dvd lrg ur sll ss sz / I ory wll rcovrd wh 1 d I d d III PARAMERIAIO h gv rrrzo s dd vrbl I h v sll ss sz hv / +1 Cosdr dd wh h couu sd 1 couu ory wllsys rcovrd wo vrbls 1 d ( l g sz l w rvl xd hr l u hr +1 ( (1 A sdrd lc rocdur d rscld oc hg or: A sdrd lc rocdur d rscld vry slr grl rlvy whch ully ls us ss 1 sys ( volvg ( voluo rr d gov- Cosdr s 0 d kg sz rvl xd orcovrd suc: w ld sys owh sd wh sco I c sco I c dlss vrbls d h xrs Idd bov rd dlss vrbls d W ( ; l d (6 vr udr rrrzo s dd Cosdr sys dd wo vrbls ( lu 1 0 sys volvg rd sys co d wh xdd sz( s Dcrz o ss sc rvl d dscrd us goh xd zro oc rucy brd or0 hs h dlss co cos h 1 dlss co srd coord 1 (D vrc d ( volvg dscrd d govo w sz (+1 Wh cos gr vrbls voluo sys ( dss sy co c s Mor rcly wwh dcrzd cou rr co lzo dlss vrbls uos ow ( +1 As L us dcrz hco sys or x rvl rd co dcrzd (+ w w o? vro co vrbls ( d c s 1 orh sys dscrd 1 s d (7 cou r lud s whr 0 d subl +1 (3 ( ( d sly ( (3 +1 sl ss sz d +1 ( d co c dcrzd s ollows sur or cr or orlzo whch gvs whch gvs ys volvg ( d ( Cosdr dcrzd +1 H( (7 rr h /d c h (lwhr od o oly kg I words dcrzd co ully do dc drvv wh rsc d (1 ; d /d +1 r co E+1 E dlss ( 1 bu l sdg whr ll us o ow whr ll us o rhs r ow dlss (1 rhs crcl ( s vlu ss h lry obsrvo whr do dc [( ( ] I d s h h hysclly ully uv +1 W ( ; D[( ]D[( ] [(] h co (or r lycl couo Eus crh co (or r s lycl couo udy Eu-I (0 +1 l dg ( D[(] ur (5 hroc whr do dc drvv wh rsc rcl L us cosucs dcrzo r d s l oscllr dcussd rvous ( h rgy ow cosrv +1 Dcrz rocdur rd lc d rscld (0 ( cld c sudd urclly gv uv (10 cld c sudd gv I d s h h hysclly ully lghs dcrzd sys cosuc c l h hro sco I c uo oo or +1 urclly cl vlu corrlo I ossbl cus dd (13 A sdrd lc rocdur d rscld lss vrbls d roxo h grl r roxo h grl l hroc oscllr dcussd rvous dvrg ur lc ss h y ory o gv doubl sco I c l gv whch o h gv ocdur d rscld dlss co cos dlss vrbls d d or whch whch o o uo oo sco I c (8 dycs ordr or oc hg or: uy dros ro s d h dlss co cos [(] ( cosrvo ( [(] cosrvo o D[(] d (4 D[(] d (4 ( ( I or words coous h xrs Idd bov rds sco ;( cos l d (6 ( (3 W ( (14 +1 ; l d d d hs hs dd dd whch 11 0 hroc (8 dgr rdo whch u gvs dly oscllr uo ( ( ( rrrz rrrz (+1l /d couu uo w o odk k couu l w hv hv sd sd 1 1whchdycs o-rrzd br ; whl or gvs d ( (3 ( (11 whr d d vry sl +1 vry s 0 h o suc: 0w us us o r bu l sd s d rhs ow d dlss whch subl bu h o suc: w us l sd sd /d gvs dly hroc oscllr uo Bu dcr rrzd ; dy wh +1 r o or l k h 1 vr udr vr ud (orcr r s lycl couod Euwhr ll ouo rhs r ow crcl us dsc ur 0 d us go zro o or sur vlu 0 sc ddlss us go ddd zro rrz ss rdo crcl ;vlu whl or yss volvg Wh : /d couu l od kg y + 0 (9 coord lrg vrg ss r ucoord vr vrd c l sudd urclly gv rhs r ow dlss h co (or r s lycl couo Eu co l Mor rcly sy w w cou h od o oly kg I words dcrzd ully dd ro l Mor rcly sy w w cou d coulg cos crcl vlu xg h rgy cosrv clly sll cld dc L L us us dcr dcr h grl o s lycl couo Eu sudd urclly gv - rogr u l sdg s crcl vlu ss- rogr h lry o hrd s obsrvo + 0 (9 Blow I sudy whr clsscl d u sl whch o dd uo: sly sl r covr Grl crcouu l kg jus ur ss y sudd urclly gv syss: roxo od us h grl rcl L sudy cosucs h c g ur dcrzo r dcr ory: y u such dcrzo ro (8 sys (dycs ( cosrvo rgy h ollows whch d o ( H( hd[(] grl H( d [(] d (4 gv dcrzd sys corrlo lghs cosuc h l W ( h djus cosrvo rgy W ( h ;; couu rrzd ory rd[(] wlldd dd d [(] sly ( hs lrg udr rbrry co o dd uo: vrc lc ss h ssbl d (4 gug ur ywhch r ory o gv doubl l 1 cosrvo co gro dcr ( ( [(] rrrzo s dd vrbl I h hs dd l rgy h ollows ro (8 sys[(] d l [(] h couu w(hv (4 sd cosrvo 1

22 All s r y oruls ul w c do hyscs wh

23 All s r y oruls ul w c do hyscs wh A cocr clculo obsrvbl: Fs Rdo Burss d blck wh hol ullg

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28 I I

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34 Covr loo uu grvy Clculo ( j -k j k A(j k 1 l

35 Covr loo uu grvy Clculo ( j -k j k A(j k 1 l

36 Covr loo uu grvy Clculo ( j -k j k A(j k 1 l

37 Covr loo uu grvy Clculo ( j -k j k A(j k 1 l

38 Covr loo uu grvy Clculo ( j -k j k A(j k 1 l

39 Covr loo uu grvy Clculo ( j -k j k jk v j-k A(j k 1 l

40 Covr loo uu grvy Clculo ( j -k j k jk v A(j k L(C dg U dh dh + U j + j (j + j (j + j r j+ [ k 3 Y gy ]r j [ k 3 Y gy ] j-k A(j k 1 l

41 Covr loo uu grvy Clculo ( j -k j k jk v A(j k L(C dg U dh dh + U j + j (j + j (j + j r j+ [ k 3 Y gy ]r j [ k 3 Y gy ] j-k A(j k 1 rlo j-k l

42 Covr loo uu grvy Clculo ( j -k j k jk v A(j k L(C dg U dh dh + U j + j (j + j (j + j r j+ [ k 3 Y gy ]r j [ k 3 Y gy ] j-k A(j k 1 rlo j-k rlo - l

43 4 Dcbl? Alrdy dcd? For 3 rordl blck hol gv sgls cosc ry scru Plck sr hoology For rordl blck hol gv sgls rdo: Fs Rdo Burss? Fs Rdo Burss d Wh Hol gls Aurl Brru Crlo Rovll PhysL B739 ( Aurél Brru Crlo Rovll Frcsc Vdo PhysRv D90 ( Fs Rdo Burss Duro: llcods Frucy: 13 GHz Fgur Chrcrc los FRB 1110 Obsrvd : Prks Arcbo d wr od d rucy rscvly l dyc sc Org: Lkly xrglcc ry obsrvo showg 07 s durg whch F Esd d owr: 1038 rg cross rucy bd sgl s cly dr wrds lowr r b Physcl urc: ukow cs du RFI r l vbl wo wh c cd sw or ν drsd sgl DM lowr l l shows ddrsd ul cross bdss lowr rgh l co scru (blck wh off-uls scru (l

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