Constraints on black-hole charges with the 2017 EHT observations of M87*
Prashant Kocherlakota,
1
Luciano Rezzolla,
1, 2, 3
Heino Falcke,
4
Christian M. Fromm,
5, 6, 1
Michael Kramer,
7
Yosuke Mizuno,
8, 9
Antonios Nathanail,
9, 10
H
́
ector Olivares,
4
Ziri Younsi,
11, 9
Kazunori Akiyama,
12, 13, 5
Antxon Alberdi,
14
Walter Alef,
7
Juan
Carlos Algaba,
15
Richard Anantua,
5, 6, 16
Keiichi Asada,
17
Rebecca Azulay,
18, 19, 7
Anne-Kathrin Baczko,
7
David Ball,
20
Mislav
Balokovi
́
c,
5, 6
John Barrett,
12
Bradford A. Benson,
21, 22
Dan Bintley,
23
Lindy Blackburn,
5, 6
Raymond Blundell,
6
Wilfred
Boland,
24
Katherine L. Bouman,
5, 6, 25
Geoffrey C. Bower,
26
Hope Boyce,
27, 28
Michael Bremer,
29
Christiaan D. Brinkerink,
4
Roger Brissenden,
5, 6
Silke Britzen,
7
Avery E. Broderick,
30, 31, 32
Dominique Broguiere,
29
Thomas Bronzwaer,
4
Do-Young
Byun,
33, 34
John E. Carlstrom,
35, 22, 36, 37
Andrew Chael,
38, 39
Chi-kwan Chan,
20, 40
Shami Chatterjee,
41
Koushik Chatterjee,
42
Ming-Tang Chen,
26
Yongjun Chen (
陈
永
军
),
43, 44
Paul M. Chesler,
5
Ilje Cho,
33, 34
Pierre Christian,
45
John E. Conway,
46
James
M. Cordes,
41
Thomas M. Crawford,
22, 35
Geoffrey B. Crew,
12
Alejandro Cruz-Osorio,
9
Yuzhu Cui,
47, 48
Jordy Davelaar,
49, 16, 4
Mariafelicia De Laurentis,
50, 9, 51
Roger Deane,
52, 53, 54
Jessica Dempsey,
23
Gregory Desvignes,
55
Sheperd S. Doeleman,
5, 6
Ralph P. Eatough,
56, 7
Joseph Farah,
6, 5, 57
Vincent L. Fish,
12
Ed Fomalont,
58
Raquel Fraga-Encinas,
4
Per Friberg,
23
H. Alyson
Ford,
59
Antonio Fuentes,
14
Peter Galison,
5, 60, 61
Charles F. Gammie,
62, 63
Roberto Garc
́
ıa,
29
Olivier Gentaz,
29
Boris
Georgiev,
31, 32
Ciriaco Goddi,
4, 64
Roman Gold,
65, 30
Jos
́
e L. G
́
omez,
14
Arturo I. G
́
omez-Ruiz,
66, 67
Minfeng Gu (
顾
敏
峰
),
43, 68
Mark Gurwell,
6
Kazuhiro Hada,
47, 48
Daryl Haggard,
27, 28
Michael H. Hecht,
12
Ronald Hesper,
69
Luis C. Ho (
何
子
山
),
70, 71
Paul
Ho,
17
Mareki Honma,
47, 48, 72
Chih-Wei L. Huang,
17
Lei Huang (
黄
磊
),
43, 68
David H. Hughes,
66
Shiro Ikeda,
13, 73, 74, 75
Makoto
Inoue,
17
Sara Issaoun,
4
David J. James,
5, 6
Buell T. Jannuzi,
20
Michael Janssen,
7
Britton Jeter,
31, 32
Wu Jiang (
江
悟
),
43
Alejandra Jimenez-Rosales,
4
Michael D. Johnson,
5, 6
Svetlana Jorstad,
76, 77
Taehyun Jung,
33, 34
Mansour Karami,
30, 31
Ramesh
Karuppusamy,
7
Tomohisa Kawashima,
78
Garrett K. Keating,
6
Mark Kettenis,
79
Dong-Jin Kim,
7
Jae-Young Kim,
33, 7
Jongsoo
Kim,
33
Junhan Kim,
20, 25
Motoki Kino,
13, 80
Jun Yi Koay,
17
Yutaro Kofuji,
47, 72
Patrick M. Koch,
17
Shoko Koyama,
17
Carsten
Kramer,
29
Thomas P. Krichbaum,
7
Cheng-Yu Kuo,
81, 17
Tod R. Lauer,
82
Sang-Sung Lee,
33
Aviad Levis,
25
Yan-Rong Li (
李
彦
荣
),
83
Zhiyuan Li (
李
志
远
),
84, 85
Michael Lindqvist,
46
Rocco Lico,
14, 7
Greg Lindahl,
6
Jun Liu (
刘
俊
),
7
Kuo Liu,
7
Elisabetta
Liuzzo,
86
Wen-Ping Lo,
17, 87
Andrei P. Lobanov,
7
Laurent Loinard,
88, 89
Colin Lonsdale,
12
Ru-Sen Lu (
路
如
森
),
43, 44, 7
Nicholas
R. MacDonald,
7
Jirong Mao (
毛
基
荣
),
90, 91, 92
Nicola Marchili,
86, 7
Sera Markoff,
42, 93
Daniel P. Marrone,
20
Alan P.
Marscher,
76
Iv
́
an Mart
́
ı-Vidal,
18, 19
Satoki Matsushita,
17
Lynn D. Matthews,
12
Lia Medeiros,
94, 20
Karl M. Menten,
7
Izumi
Mizuno,
23
James M. Moran,
5, 6
Kotaro Moriyama,
12, 47
Monika Moscibrodzka,
4
Cornelia M
̈
uller,
7, 4
Gibwa Musoke,
42, 4
Alejandro Mus Mej
́
ıas,
18, 19
Hiroshi Nagai,
13, 48
Neil M. Nagar,
95
Masanori Nakamura,
96, 17
Ramesh Narayan,
5, 6
Gopal
Narayanan,
97
Iniyan Natarajan,
54, 52, 98
Joseph Neilsen,
99
Roberto Neri,
29
Chunchong Ni,
31, 32
Aristeidis Noutsos,
7
Michael A.
Nowak,
100
Hiroki Okino,
47, 72
Gisela N. Ortiz-Le
́
on,
7
Tomoaki Oyama,
47
Feryal
̈
Ozel,
20
Daniel C. M. Palumbo,
5, 6
Jongho
Park,
17
Nimesh Patel,
6
Ue-Li Pen,
30, 101, 102, 103
Dominic W. Pesce,
5, 6
Vincent Pi
́
etu,
29
Richard Plambeck,
104
Aleksandar
PopStefanija,
97
Oliver Porth,
42, 9
Felix M. P
̈
otzl,
7
Ben Prather,
62
Jorge A. Preciado-L
́
opez,
30
Dimitrios Psaltis,
20
Hung-Yi
Pu,
105, 17, 30
Venkatessh Ramakrishnan,
95
Ramprasad Rao,
26
Mark G. Rawlings,
23
Alexander W. Raymond,
5, 6
Angelo
Ricarte,
5, 6
Bart Ripperda,
106, 16
Freek Roelofs,
4
Alan Rogers,
12
Eduardo Ros,
7
Mel Rose,
20
Arash Roshanineshat,
20
Helge
Rottmann,
7
Alan L. Roy,
7
Chet Ruszczyk,
12
Kazi L. J. Rygl,
86
Salvador S
́
anchez,
107
David S
́
anchez-Arguelles,
66, 67
Mahito
Sasada,
47, 108
Tuomas Savolainen,
109, 110, 7
F. Peter Schloerb,
97
Karl-Friedrich Schuster,
29
Lijing Shao,
7, 71
Zhiqiang Shen
(
沈
志强
),
43, 44
Des Small,
79
Bong Won Sohn,
33, 34, 111
Jason SooHoo,
12
He Sun (
孙
赫
),
25
Fumie Tazaki,
47
Alexandra
J. Tetarenko,
112
Paul Tiede,
31, 32
Remo P. J. Tilanus,
4, 64, 113, 20
Michael Titus,
12
Kenji Toma,
114, 115
Pablo Torne,
7, 107
Tyler Trent,
20
Efthalia Traianou,
7
Sascha Trippe,
116
Ilse van Bemmel,
79
Huib Jan van Langevelde,
79, 117
Daniel R. van
Rossum,
4
Jan Wagner,
7
Derek Ward-Thompson,
118
John Wardle,
119
Jonathan Weintroub,
5, 6
Norbert Wex,
7
Robert
Wharton,
7
Maciek Wielgus,
5, 6
George N. Wong,
62
Qingwen Wu (
吴
庆
文
),
120
Doosoo Yoon,
42
Andr
́
e Young,
4
Ken Young,
6
Feng Yuan (
袁
峰
),
43, 68, 121
Ye-Fei Yuan (
袁
业
飞
),
122
J. Anton Zensus,
7
Guang-Yao Zhao,
14
and Shan-Shan Zhao
43
(The EHT Collaboration)
1
Institut f
̈
ur Theoretische Physik, Goethe-Universit
̈
at, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany
2
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt, Germany
3
School of Mathematics, Trinity College, Dublin 2, Ireland
4
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP),
Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
5
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
6
Center for Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
7
Max-Planck-Institut f
̈
ur Radioastronomie, Auf dem H
̈
ugel 69, D-53121 Bonn, Germany
8
Tsung-Dao Lee Institute and School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai, 200240, China
9
Institut f
̈
ur Theoretische Physik, Goethe-Universit
̈
at Frankfurt,
Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
arXiv:2105.09343v1 [gr-qc] 19 May 2021
2
10
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, GR 15783 Zografos, Greece
11
Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
12
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA
13
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
14
Instituto de Astrof
́
ısica de Andaluc
́
ıa-CSIC, Glorieta de la Astronom
́
ıa s/n, E-18008 Granada, Spain
15
Department of Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia
16
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
17
Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building,
AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.
18
Departament d’Astronomia i Astrof
́
ısica, Universitat de Val
`
encia,
C. Dr. Moliner 50, E-46100 Burjassot, Val
`
encia, Spain
19
Observatori Astron
`
omic, Universitat de Val
`
encia, C. Catedr
́
atico Jos
́
e Beltr
́
an 2, E-46980 Paterna, Val
`
encia, Spain
20
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA
21
Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL, 60510, USA
22
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
23
East Asian Observatory, 660 N. A’ohoku Place, Hilo, HI 96720, USA
24
Nederlandse Onderzoekschool voor Astronomie (NOVA), PO Box 9513, 2300 RA Leiden, The Netherlands
25
California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA
26
Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A’ohoku Place, Hilo, HI 96720, USA
27
Department of Physics, McGill University, 3600 rue University, Montr
́
eal, QC H3A 2T8, Canada
28
McGill Space Institute, McGill University, 3550 rue University, Montr
́
eal, QC H3A 2A7, Canada
29
Institut de Radioastronomie Millim
́
etrique, 300 rue de la Piscine, F-38406 Saint Martin d’H
`
eres, France
30
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
31
Department of Physics and Astronomy, University of Waterloo,
200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
32
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
33
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
34
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea
35
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
36
Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, IL, 60637, USA
37
Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
38
Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA
39
NASA Hubble Fellowship Program, Einstein Fellow
40
Data Science Institute, University of Arizona, 1230 N. Cherry Ave., Tucson, AZ 85721, USA
41
Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA
42
Anton Pannekoek Institute for Astronomy, University of Amsterdam,
Science Park 904, 1098 XH, Amsterdam, The Netherlands
43
Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, People’s Republic of China
44
Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People’s Republic of China
45
Physics Department, Fairfield University, 1073 North Benson Road, Fairfield, CT 06824, USA
46
Department of Space, Earth and Environment, Chalmers University of Technology,
Onsala Space Observatory, SE-43992 Onsala, Sweden
47
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan,
2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
48
Department of Astronomical Science, The Graduate University for Advanced
Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
49
Department of Astronomy and Columbia Astrophysics Laboratory,
Columbia University, 550 W 120th Street, New York, NY 10027, USA
50
Dipartimento di Fisica “E. Pancini”, Universit
́
a di Napoli “Federico II”,
Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
51
INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
52
Wits Centre for Astrophysics, University of the Witwatersrand,
1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa
53
Department of Physics, University of Pretoria, Hatfield, Pretoria 0028, South Africa
54
Centre for Radio Astronomy Techniques and Technologies,
Department of Physics and Electronics, Rhodes University, Makhanda 6140, South Africa
55
LESIA, Observatoire de Paris, Universit
́
e PSL, CNRS, Sorbonne Universit
́
e,
Universit
́
e de Paris, 5 place Jules Janssen, 92195 Meudon, France
56
National Astronomical Observatories, Chinese Academy of Sciences,
20A Datun Road, Chaoyang District, Beijing 100101, PR China
57
University of Massachusetts Boston, 100 William T. Morrissey Boulevard, Boston, MA 02125, USA
58
National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA 22903, USA
59
Steward Observatory and Department of Astronomy, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
3
60
Department of History of Science, Harvard University, Cambridge, MA 02138, USA
61
Department of Physics, Harvard University, Cambridge, MA 02138, USA
62
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA
63
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA
64
Leiden Observatory—Allegro, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
65
CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
66
Instituto Nacional de Astrof
́
ısica,
́
Optica y Electr
́
onica. Apartado Postal 51 y 216, 72000. Puebla Pue., M
́
exico
67
Consejo Nacional de Ciencia y Tecnolog
́
ıa, Av. Insurgentes Sur 1582, 03940, Ciudad de M
́
exico, M
́
exico
68
Key Laboratory for Research in Galaxies and Cosmology,
Chinese Academy of Sciences, Shanghai 200030, People’s Republic of China
69
NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute,
University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands
70
Department of Astronomy, School of Physics, Peking University, Beijing 100871, People’s Republic of China
71
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People’s Republic of China
72
Department of Astronomy, Graduate School of Science,
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
73
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan
74
Department of Statistical Science, The Graduate University for Advanced
Studies (SOKENDAI), 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
75
Kavli Institute for the Physics and Mathematics of the Universe,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
76
Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA
77
Astronomical Institute, St. Petersburg University, Universitetskij pr., 28, Petrodvorets,198504 St.Petersburg, Russia
78
Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
79
Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
80
Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan
81
Physics Department, National Sun Yat-Sen University,
No. 70, Lien-Hai Rd, Kaosiung City 80424, Taiwan, R.O.C
82
National Optical Astronomy Observatory, 950 North Cherry Ave., Tucson, AZ 85719, USA
83
Key Laboratory for Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan District, Beijing, People’s Republic of China
84
School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China
85
Key Laboratory of Modern Astronomy and Astrophysics,
Nanjing University, Nanjing 210023, People’s Republic of China
86
Italian ALMA Regional Centre, INAF-Istituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy
87
Department of Physics, National Taiwan University, No.1, Sect.4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C
88
Instituto de Radioastronom
́
ıa y Astrof
́
ısica, Universidad Nacional Aut
́
onoma de M
́
exico, Morelia 58089, M
́
exico
89
Instituto de Astronom
́
ıa, Universidad Nacional Aut
́
onoma de M
́
exico, CdMx 04510, M
́
exico
90
Yunnan Observatories, Chinese Academy of Sciences,
650011 Kunming, Yunnan Province, People’s Republic of China
91
Center for Astronomical Mega-Science, Chinese Academy of Sciences,
20A Datun Road, Chaoyang District, Beijing, 100012, People’s Republic of China
92
Key Laboratory for the Structure and Evolution of Celestial Objects,
Chinese Academy of Sciences, 650011 Kunming, People’s Republic of China
93
Gravitation Astroparticle Physics Amsterdam (GRAPPA) Institute,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
94
School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
95
Astronomy Department, Universidad de Concepci
́
on, Casilla 160-C, Concepci
́
on, Chile
96
National Institute of Technology, Hachinohe College,
16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan
97
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA
98
South African Radio Astronomy Observatory, Observatory 7925, Cape Town, South Africa
99
Villanova University, Mendel Science Center Rm. 263B, 800 E Lancaster Ave, Villanova PA 19085
100
Physics Department, Washington University CB 1105, St Louis, MO 63130, USA
101
Canadian Institute for Theoretical Astrophysics, University of Toronto,
60 St. George Street, Toronto, ON M5S 3H8, Canada
102
Dunlap Institute for Astronomy and Astrophysics, University of Toronto,
50 St. George Street, Toronto, ON M5S 3H4, Canada
103
Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, ON M5G 1Z8, Canada
104
Radio Astronomy Laboratory, University of California, Berkeley, CA 94720, USA
105
Department of Physics, National Taiwan Normal University,
No. 88, Sec.4, Tingzhou Rd., Taipei 116, Taiwan, R.O.C.
106
Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
4
107
Instituto de Radioastronom
́
ıa Milim
́
etrica, IRAM, Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain
108
Hiroshima Astrophysical Science Center, Hiroshima University,
1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
109
Aalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Aalto, Finland
110
Aalto University Mets
̈
ahovi Radio Observatory, Mets
̈
ahovintie 114, FI-02540 Kylm
̈
al
̈
a, Finland
111
Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu, 03722 Seoul, Republic of Korea
112
East Asian Observatory, 660 North A’ohoku Place, Hilo, HI 96720, USA
113
Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands
114
Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan
115
Astronomical Institute, Tohoku University, Sendai 980-8578, Japan
116
Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea
117
Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands
118
Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK
119
Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA
120
School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China
121
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences,
No. 19A Yuquan Road, Beijing 100049, People’s Republic of China
122
Astronomy Department, University of Science and Technology of China, Hefei 230026, People’s Republic of China
Our understanding of strong gravity near supermassive compact objects has recently improved thanks to
the measurements made by the Event Horizon Telescope (EHT). We use here the M87* shadow size to infer
constraints on the physical charges of a large variety of nonrotating or rotating black holes. For example, we
show that the quality of the measurements is already sufficient to rule out that M87* is a highly charged dilaton
black hole. Similarly, when considering black holes with two physical and independent charges, we are able to
exclude considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes.
I. INTRODUCTION
General relativity (GR) was formulated to consistently ac-
count for the interaction of dynamical gravitational fields with
matter and energy, the central idea of which is that the former
manifests itself through modifications of spacetime geome-
try and is fully characterized by a metric tensor. While the
physical axioms that GR is founded on are contained in the
equivalence principle [1, 2], the Einstein-Hilbert action fur-
ther postulates that the associated equations of motion involve
no more than second-order derivatives of the metric tensor.
The strength of the gravitational field outside an object
of mass
M
and characteristic size
R
, in geometrized units
(
G
=
c
= 1
), is related to its compactness
C
:=
M/R
,
which is
∼
10
−
6
for the Sun, and takes values
∼
0
.
2
−
1
for
compact objects such as neutron stars and black holes. Pre-
dictions from GR have been tested and validated by various
solar-system experiments to very high precision [2, 3], set-
ting it on firm footing as the best-tested theory of classical
gravity in the weak-field regime. It is important, however, to
consider whether signatures of deviations from the Einstein-
Hilbert action, e.g., due to higher derivative terms [4], could
appear in measurements of phenomena occurring in strong-
field regimes where
C
is large. Similarly, tests are needed to
assess whether generic violations of the equivalence principle
occur in strong-fields due, e.g., to the presence of additional
dynamical fields, such as scalar [5] or vector fields [6], that
may fall off asymptotically. Agreement with the predictions
of GR coming from observations of binary pulsars [7], and of
the gravitational redshift [8] and geodetic orbit-precession [9]
of the star S2 near our galaxy’s central supermassive compact
object Sgr A
?
by the GRAVITY collaboration, all indicate the
success of GR in describing strong-field physics as well. In
addition, with the gravitational-wave detections of coalescing
binaries of compact objects by the LIGO/Virgo collaboration
[10] and the first images of black holes produced by EHT, it
is now possible to envision testing GR at the strongest field
strengths possible.
While the inferred size of the shadow from the recently ob-
tained horizon-scale images of the supermassive compact ob-
ject in M87 galaxy by the EHT collaboration [11] was found
to be consistent to within 17% for a 68% confidence interval
of the size predicted from GR for a Schwarzschild black hole
using the
a priori
known estimates for the mass and distance
of M87* based on stellar dynamics [12], this measurement ad-
mits other possibilities, as do various weak-field tests [2, 13].
Since the number of alternative theories to be tested using this
measurement is large, a systematic study of the constraints
set by a strong-field measurement is naturally more tractable
within a theory-agnostic framework, and various such systems
have recently been explored [14, 15]. This approach allows
for tests of a broad range of possibilities that may not be cap-
tured in the limited set of known solutions. This was exploited
in Ref. [13], where constraints on two deformed metrics were
obtained when determining how different M87* could be from
a Kerr black hole while remaining consistent with the EHT
measurements.
However, because such parametric tests cannot be con-
nected directly to an underlying property of the alternative
theory, here we use instead the EHT measurements to set
bounds on the physical parameters, i.e., angular momentum,
electric charge, scalar charge, etc., – and which we will gener-
ically refer to as “charges” (or hairs) – that various well-
known black-hole solutions depend upon. Such analyses can
be very instructive [16–18] since they can shed light on which
underlying theories are promising candidates and which must
be discarded or modified. At the same time, they may provide
insight into the types of additional dynamical fields that may
5
be necessary for a complete theoretical description of physical
phenomena, and whether associated violations of the equiva-
lence principle occur.
More specifically, since the bending of light in the presence
of curvature – either in static or in stationary spacetimes –
is assured in any metric theory of gravity, and the presence of
large amounts of mass in very small volumes can allow for the
existence of a region where null geodesics move on spherical
orbits, an examination of the characteristics of such photon
regions, when they exist, is a useful first step. The projected
asymptotic collection of the photons trajectories that are cap-
tured by the black hole – namely, all of the photon trajectories
falling within the value of the impact parameter at the unsta-
ble circular orbit in the case of nonrotating black holes – will
appear as a dark area to a distant observer and thus represents
the “shadow” of the capturing compact object. This shadow
– which can obviously be associated with black holes [19–
24], but also more exotic compact objects such as gravastars
[25, 26] or naked singularities [27, 28] – is determined entirely
by the underlying spacetime metric. Therefore, the properties
of the shadow – and at lowest order its size – represent valu-
able observables common to all metric theories of gravity, and
can be used to test them for their agreement with EHT mea-
surements.
While the EHT measurement contains far more information
related to the flow of magnetised plasma near M87*, we will
consider only the measurement of the size of the bright ring.
Here we consider various spherically symmetric black-hole
solutions, from GR that are either singular (see, e.g., [29]) or
non-singular [30–32], and string theory [33–37]. Addition-
ally, we also consider the Reissner-Nordstr
̈
om (RN) and the
Janis-Newman-Winicour (JNW) [38] naked singularity solu-
tions, the latter being a solution of the Einstein-Klein-Gordon
system. Many of these solutions have been recently sum-
marised in Ref. [15], where they were cast in a generalised ex-
pansion of static and spherically symmetric metrics. Since an-
gular momentum plays a key role in astrophysical scenarios,
we also consider various rotating black-hole solutions [39–
42] which can be expressed in the Newman-Janis form [43] to
facilitate straightforward analytical computations. It is to be
noted that this study is meant to be a proof of principle and
that while the constraints we can set here are limited, the ana-
lytical procedure outlined below for this large class of metrics
is general, so that as future observations become available, we
expect the constraints that can be imposed following the ap-
proach proposed here to be much stronger.
II. SPHERICAL NULL GEODESICS AND SHADOWS
For all the static, spherically symmetric spacetimes we con-
sider here, the definition of the shadow can be cast in rather
general terms. In particular, for all the solutions considered,
the line element expressed in areal-radial polar coordinates
(
t,
̃
r,θ,φ
)
has the form
1
ds
2
=
g
μν
dx
μ
dx
ν
=
−
f
( ̃
r
)
d
t
2
+
g
( ̃
r
)
f
( ̃
r
)
d
̃
r
2
+ ̃
r
2
d
Ω
2
2
,
(1)
and the photon region, which degenerates into a photon
sphere, is located at
̃
r
=: ̃
r
ps
, which can be obtained by solv-
ing [13]
̃
r
−
2
f
( ̃
r
)
∂
̃
r
f
( ̃
r
)
= 0
.
(2)
The boundary of this photon sphere when observed from the
frame of an asymptotic observer, due to gravitational lensing,
appears to be a circle of size [13]
̃
r
sh
=
̃
r
ps
√
f
( ̃
r
ps
)
.
(3)
On the other hand, the class of Newman-Janis stationary,
axisymmetric spacetimes we consider here [43], which are
geodesically integrable (see, e.g., [22, 44, 45]), can be ex-
pressed in Boyer-Lindquist coordinates (
t,r,θ,φ
) as
d
s
2
=
−
f
d
t
2
−
2
a
sin
2
θ
(1
−
f
)
d
t
d
φ
(4)
+
[
Σ +
a
2
sin
2
θ
(2
−
f
)
]
sin
2
θ
d
φ
2
+
Σ
∆
d
r
2
+ Σ
d
θ
2
,
where
f
=
f
(
r,θ
)
and
Σ(
r,θ
) :=
r
2
+
a
2
cos
2
θ
and
∆(
r
) :=
Σ(
r,θ
)
f
(
r,θ
) +
a
2
sin
2
θ
. In particular, these are the sta-
tionary generalisations obtained by employing the Newman-
Janis algorithm [43]) for “seed” metrics of the form (1) with
g
( ̃
r
) = 1
2
.
The Lagrangian
L
for geodesic motion in the spacetime (4)
is given as
2
L
:=
g
μν
̇
x
μ
̇
x
ν
, where an overdot represents a
derivative with respect to the affine parameter, and
2
L
=
−
1
for timelike geodesics and
2
L
= 0
for null geodesics. The
two Killing vectors
∂
t
and
∂
φ
yield two constants of motion
−
E
=
−
f
̇
t
−
a
sin
2
θ
(1
−
f
)
̇
φ,
(5)
L
=
−
a
sin
2
θ
(1
−
f
)
̇
t
+
[
Σ +
a
2
sin
2
θ
(2
−
f
)
]
sin
2
θ
̇
φ,
in terms of which the geodesic equation for photons can be
separated into
Σ
2
̇
r
2
= (
r
2
+
a
2
−
aξ
)
2
−
∆
I
=:
R
(
r
)
,
(6)
Σ
2
̇
θ
2
=
I−
(
a
sin
θ
−
ξ
csc
θ
)
2
=: Θ(
θ
)
,
(7)
where we have introduced first
ξ
:=
L/E
, and then
I
:=
η
+ (
a
−
ξ
)
2
. Also,
η
is the Carter constant, and the exis-
tence of this fourth constant of motion is typically associated
1
We use the tilde on the radial coordinate of static spacetimes to distinguish
it from the corresponding radial coordinate of axisymmetric spacetimes.
2
Note that while the Sen solution can be obtained via the Newman-Janis
algorithm [46], the starting point is the static EMd-1 metric written in a
non
-areal-radial coordinate
ρ
such that
g
tt
g
ρρ
=
−
1
.
6
with the existence of an additional Killing-Yano tensor (see
for example [23, 47]).
In particular, we are interested here in spherical null
geodesics (SNGs), which satisfy
̇
r
= 0
and
̈
r
= 0
and are
not necessarily planar; equivalently, SNGs can exist at loca-
tions where
R
(
r
) = 0
and d
R
(
r
)
/
d
r
= 0
. Since these are
only two equations in three variables (
r,ξ,η
), it is convenient,
for reasons that will become evident below, to obtain the asso-
ciated conserved quantities along such SNGs in terms of their
radii
r
as (see also [48]),
ξ
SNG
(
r
) =
r
2
+
a
2
a
−
4
r
∆
a∂
r
∆
,
(8)
η
SNG
(
r
) =
r
2
a
2
(
∂
r
∆)
2
[
16
a
2
∆
−
(
r∂
r
∆
−
4∆)
2
]
.
The condition that
Θ(
θ
)
≥
0
, which must necessarily hold as
can be seen from Eq. (7), restricts the radial range for which
SNGs exist, and it is evident that this range depends on
θ
.
This region, which is filled by such SNGs, is called the photon
region (see, e.g., Fig. 3.3 of [19]).
The equality
Θ(
θ
) = 0
determines the boundaries of the
photon region, and the (disconnected) piece which lies in the
exterior of the outermost horizon is of primary interest since
its image, as seen by an asymptotic observer, is the shadow.
We denote the inner and outer surfaces of this photon region
by
r
p
−
(
θ
)
and
r
p
+
(
θ
)
respectively, with the former (smaller)
SNG corresponding to the location of a prograde photon orbit
(i.e.,
ξ
SNG
(
r
p
−
)
>
0
), and the latter to a retrograde orbit.
It can be shown that all of the SNGs that are admitted in
the photon region, for both the spherically symmetric and ax-
isymmetric solutions considered here, are unstable to radial
perturbations. In particular, for the stationary solutions, the
stability of SNGs at a radius
r
=
r
SNG
with respect to radial
perturbations is determined by the sign of
∂
2
r
R
, and when
∂
2
r
R
(
r
SNG
)
>
0
, SNGs at that radius are unstable. The ex-
pression for
∂
2
r
R
reads
∂
2
r
R
=
8
r
(
∂
r
∆)
2
[
r
(
∂
r
∆)
2
−
2
r
∆
∂
2
r
∆ + 2∆
∂
r
∆
]
.
(9)
To determine the appearance of the photon region and the as-
sociated shadow, as seen by asymptotic observers, we can in-
troduce the usual notion of celestial coordinates
(
α,β
)
, which
for any photon with constants of the motion
(
ξ,η
)
can be ob-
tained, for an asymptotic observer present at an inclination
angle
i
with respect to the spin-axis of the compact object as
in [49]. For photons on an SNG, we can set the conserved
quantities (
ξ,η
) to the values given in Eq. 8 above to obtain
[47, 48]
α
sh
=
−
ξ
SNG
csc
i,
(10)
β
sh
=
±
(
η
SNG
+
a
2
cos
2
i
−
ξ
2
SNG
cot
2
i
)
1
/
2
.
(11)
Recognizing that
β
=
±
√
Θ(
i
)
, it becomes clear that only the
SNGs with
Θ(
i
)
≥
0
determine the apparent shadow shape.
Since the photon region is not spherically symmetric in ro-
tating spacetimes, the associated shadow is also not circular
in general. It can be shown that the band of radii for which
SNGs can exist narrows as we move away from the equatorial
plane, and reduces to a single value at the pole, i.e. in the limit
θ
→
π/
2
, we have
r
p
+
=
r
p
−
(see e.g., Fig. 3.3 of [19]). As
a result, the parametric curve of the shadow boundary as seen
by an asymptotic observer lying along the pole is perfectly
circular,
α
2
sh
+
β
2
sh
=
η
SNG
(
r
p
±
,π/
2
) +
ξ
2
SNG
(
r
p
±
,π/
2
)
.
We can now define the characteristic areal-radius of the
shadow curve as [50]
r
sh
,A
:=
(
2
π
∫
r
p
+
r
p
−
d
r β
sh
(
r
)
∂
r
α
sh
(
r
)
)
1
/
2
.
(12)
III. SHADOW SIZE CONSTRAINTS FROM THE 2017 EHT
OBSERVATIONS OF M87*
Measurements of stellar dynamics near M87* were previ-
ously used to produce a posterior distribution function of the
angular gravitational radius
θ
g
:=
M/D
, where
M
is the mass
of and
D
the distance to M87*. The 2017 EHT observations
of M87* can be similarly used to determine such a posterior
[11]. These observations were used to determine the angu-
lar diameter
ˆ
d
of the bright emission ring that surrounds the
shadow [11]. In Sec. 5.3 there, using synthetic images from
general-relativistic magnetohydrodynamics (GRMHD) simu-
lations of accreting Kerr black holes for a wide range of phys-
ical scenarios, the scaling factor
α
=
ˆ
d/θ
g
was calibrated. For
emission from the outermost boundary of the photon region of
a Kerr black hole,
α
should lie in the range
'
9
.
6
−
10
.
4
.
The EHT measurement picks out a class of best-fit images
(“top-set”) from the image library, with a mean value for
α
of
11
.
55
(for the “xs-ring” model) and
11
.
50
(for the “xs-
ringauss” model), when using two different geometric cres-
cent models for the images, implying that the geometric mod-
els were accounting for emission in the top-set GRMHD im-
ages that preferentially fell outside of the photon ring. Us-
ing the distribution of
α
for these top-set images then enabled
the determination of the posterior in the angular gravitational
radius
P
obs
(
θ
g
)
for the EHT data. It is to be noted that this
posterior was also determined using direct GRMHD fitting,
and image domain feature extraction procedures, as described
in Sec. 9.2 there, and a high level of consistency was found
across all measurement methods. Finally, in Sec. 9.5 of [11],
the fractional deviation in the angular gravitational radius
δ
was introduced as
δ
:=
θ
g
θ
dyn
−
1
,
(13)
where
θ
g
and
θ
dyn
were used to denote the EHT and the stellar-
dynamics inferences of the angular gravitational radius, re-
spectively. The posterior on
δ
– as defined in Eq. (32) of [11]
– was then obtained (see Fig. 21 there), and its width was
found to be
δ
=
−
0
.
01
±
0
.
17
, for a 68% credible interval.
This agreement of the 2017 EHT measurement of the angu-
lar gravitational radius for M87* with a previously existing
estimate for the same, at much larger distances, constitutes a
7
validation of the null hypothesis of the EHT, and in particular
that M87* can be described by the Kerr black-hole solution.
Since the stellar dynamics measurements [12] are sen-
sitive only to the monopole of the metric (i.e., the mass)
due to negligible spin-dependent effects at the distances in-
volved in that analysis, modeling M87* conservatively using
the Schwarzschild solution is reasonable with their obtained
posterior. Then, using the angular gravitational radius esti-
mate from stellar dynamics yields a prediction for the angular
shadow radius
θ
sh
=
r
sh
/D
as being
θ
sh
= 3
√
3
θ
dyn
. The 2017
EHT measurement, which includes spin-dependent effects as
described above and which probes near-horizon scales, then
determines the allowed spread in the angular shadow diame-
ter as,
θ
sh
≈
3
√
3(1
±
0
.
17)
θ
g
, at 68% confidence levels [13].
Finally, since both angular estimates
θ
sh
and
θ
g
make use of
the same distance estimate to M87*, it is possible to convert
the
1
-
σ
bounds on
θ
sh
to bounds on the allowed shadow size
for M87*.
That is, independently of whether the underlying solution
be spherically symmetric (in which case we will consider
̃
r
sh
)
or axisymmetric
(
r
sh
,A
)
, the shadow size of M87* must lie in
the range
3
√
3(1
±
0
.
17)
M
[13], i.e., (see gray-shaded region
in Fig. 2)
4
.
31
M
≈
r
sh, EHT-min
≤
̃
r
sh
,r
sh
,A
≤
r
sh, EHT-max
≈
6
.
08
M ,
(14)
where we have introduced the maximum/minimum shadow
radii
r
sh, EHT-max
/r
sh, EHT-min
inferred by the EHT, at 68% con-
fidence levels.
Note that the bounds thus derived are consistent with com-
pact objects that cast shadows that are both significantly
smaller and larger than the minimum and maximum shadow
sizes that a Kerr black hole could cast, which lie in the range,
4
.
83
M
−
5
.
20
M
(see, e.g., [13, 51]).
An important caveat here is that the EHT posterior distribu-
tion on
θ
g
was obtained after a comparison with a large library
of synthetic images built from GRMHD simulations of accret-
ing Kerr black holes [11]. Ideally, a rigorous comparison with
non-Kerr solutions would require a similar analysis and poste-
rior distributions built from equivalent libraries obtained from
GRMHD simulations of such non-Kerr solutions. Besides be-
ing computationally unfeasible, this approach is arguably not
necessary in practice. For example, the recent comparative
analysis of Ref. [17] has shown that the image libraries pro-
duced in this way would be very similar and essentially indis-
tinguishable, given the present quality of the observations. As
a result, we adopt here the working assumption that the
1
-
σ
uncertainty in the shadow angular size for non-Kerr solutions
is very similar to that for Kerr black holes, and hence employ
the constraints (14) for all of the solutions considered here.
IV. NOTABLE PROPERTIES OF VARIOUS SPACETIMES
As mentioned above, a rigorous comparison with non-Kerr
black holes would require constructing a series of exhaustive
libraries of synthetic images obtained from GRMHD simu-
lations on such non-Kerr black holes. In turn, this would
TABLE I. Summary of properties of spacetimes used here. For easy
access, we show whether the spacetime contains a rotating compact
object or not, whether it contains a spacetime singularity, and what
type of stationary nongravitational fields are present in the spacetime.
Starred spacetimes contain naked singularities and daggers indicate
a violation of the equivalence principle (see, e.g., [15]); In particular,
these indicate violations of the weak equivalence principle due to a
varying fine structure constant, a result of the coupling of the dilaton
to the EM Lagrangian [15, 52].
Spacetime
Rotation
Singularity
Spacetime content
KN [40]
Yes
Yes
EM fields
Kerr [39]
Yes
Yes
vacuum
RN [29]
No
Yes
EM fields
RN* [29]
No
Yes
EM fields
Schwarzschild [29]
No
Yes
vacuum
Rot. Bardeen [42]
Yes
No
matter
Bardeen [30]
No
No
matter
Rot. Hayward [42]
Yes
No
matter
Frolov [32]
No
No
EM fields, matter
Hayward [31]
No
No
matter
JNW* [38]
No
Yes
scalar field
KS [33]
No
Yes
vacuum
Sen
†
[41]
Yes
Yes
EM, dilaton, axion fields
EMd-1
†
[34, 35]
No
Yes
EM, dilaton fields
EMd-2
†
[37]
No
Yes
EM, EM, dilaton fields
provide consistent posterior distributions of angular gravita-
tional radii for the various black holes and hence determine
how
δ
varies across different non-Kerr black holes, e.g., for
Sen black holes. Because this is computationally unfeasible –
the construction of only the Kerr library has required the joint
effort of several groups with the EHTC over a good fraction
of a year – we briefly discuss below qualitative arguments to
support our use of the bounds given in Eq. 14 above as an
approximate, yet indicative, measure.
To this end, we summarize in Table I the relevant properties
of the various solutions used here. First, we have considered
here solutions from three types of theories, i.e., the under-
lying actions are either (a) Einstein-Hilbert-Maxwell-matter
[29–33, 38, 39, 42], (b) Einstein-Hilbert-Maxwell-dilaton-
axion [34, 35, 41], or (c) Einstein-Hilbert-Maxwell-Maxwell-
dilaton [37]. This careful choice implies that the gravitational
piece of the action is always given by the Einstein-Hilbert
term and that matter is minimally coupled to gravity. As a
result, the dynamical evolution of the accreting plasma is ex-
pected to be very similar to that in GR, as indeed found in Ref.
[17]. Second, since a microphysical description that allows
one to describe the interaction of the exotic matter present in
some of the regular black-hole spacetimes used here [30, 31]
– which typically do not satisfy some form of the energy con-
ditions [42, 53] – with the ordinary matter is thus far lacking,
8
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
normalized physical charge
1
.
0
1
.
5
2
.
0
2
.
5
3
.
0
3
.
5
4
.
0
̃
r
ps
[
M
]
Schwarzschild
RN
RN
∗
Kerr (ret
.
)
Kerr (pro
.
)
Bardeen
Hayward
JNW
∗
EMd-1
KS
Schwarzschild
RN
RN
∗
Kerr (ret
.
)
Kerr (pro
.
)
Bardeen
Hayward
JNW
∗
EMd-1
KS
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
normalized physical charge
2
3
4
5
6
7
8
9
̃
r
ISCO
[
M
]
Schwarzschild
RN
RN
∗
Kerr (ret
.
)
Kerr (pro
.
)
Bardeen
Hayward
JNW
∗
EMd-1
KS
Schwarzschild
RN
RN
∗
Kerr (ret
.
)
Kerr (pro
.
)
Bardeen
Hayward
JNW
∗
EMd-1
KS
FIG. 1.
Left:
variation in the photon sphere radii for the single-charge nonrotating solutions as a function of the normalized physical charge.
Right:
The same as in the left panel but for the ISCO radii. We include also, for comparison, the variation in the Kerr equatorial prograde and
retrograde photon sphere and ISCO radii in the left and right panels respectively.
it is reasonable to assume that the interaction between these
two types of fluids is gravitational only. This is indeed what
is done in standard numerical simulations, either in dynamical
spacetimes (see, e.g., [54]), or in fixed ones [16, 55]. Third,
since the mass-energy in the matter and electromagnetic fields
for the non-vacuum spacetimes used here is of the order of the
mass of the central compact object
M
, while the total mass
of the accreting plasma in the GRMHD simulations is only a
tiny fraction of the same, it is reasonable to treat the space-
time geometry and the stationary fields as unaffected by the
plasma. Fourth, we have also been careful not to use solutions
from theories with modified electrodynamics (such as non-
linear electrodynamics). As a result, the electromagnetic La-
grangian in all of the theories considered here is the Maxwell
Lagrangian (see, e.g., the discussion in [15] and compare with
[20]). This ensures that in these spacetimes light moves along
the null geodesics of the metric tensor (see, e.g., Sec. 4.3 of
[29] and compare against Sec. 2 of [56]). Therefore, we are
also assured that ray-tracing the radiation emitted from the ac-
creting matter in these spacetimes can be handled similarly as
in the Kerr spacetime.
Finally, under the assumption that the dominant effects in
determining the angular gravitational radii come from varia-
tions in the location of the photon region and in location of the
inner edge of the accretion disk in these spacetimes, it is in-
structive to learn how these two physical quantities vary when
changing physical charges, and, in particular, to demonstrate
that they are quantitatively comparable to the corresponding
values for the Kerr spacetime.
For this purpose, we study the single-charge solutions used
here and report in Fig. 1 the variation in the location of the
photon spheres (left panel) and innermost stable circular orbit
(ISCO) radii (right panel) as a function of the relevant physical
charge (cf. left panel of Fig. 1 in the main text). Note that both
the photon-sphere and the ISCO radii depend exclusively on
the
g
tt
component of the metric when expressed using an areal
radial coordinate
̃
r
(see, e.g., [13, 15]). To gauge the effect of
spin, we also show the variation in the locations of the equa-
torial prograde and retrograde circular photon orbits and the
ISCOs in the Kerr black-hole spacetime, expressed in terms
of the Cartesian Kerr-Schild radial coordinate
r
CKS
, which, in
the equatorial plane, is related to the Boyer-Lindquist radial
coordinate used elsewhere in this work
r
simply via [57]
r
CKS
=
√
r
2
+
a
2
.
(15)
It is apparent from Fig. 1 that the maximum deviation in the
photon-sphere size from the Schwarzschild solution occurs for
the EMd-1 black hole and is
≈
75%
, while the size of the pro-
grade equatorial circular photon orbit for Kerr deviates by at
most
≈
50%
. Similarly, the maximum variation in the ISCO
size also occurs for the EMd-1 solution and is
≈
73%
, while
the prograde equatorial ISCO for Kerr can differ by
≈
66%
.
V. CHARGE CONSTRAINTS FROM THE EHT M87*
OBSERVATIONS
We first consider compact objects with a single “charge,”
and report in the left panel of Fig. 2 the variation in the
shadow radius for various spherically symmetric black hole
solutions, as well as for the RN and JNW naked singularities
3
.
More specifically, we consider the black-hole solutions given
by Reissner-Nordstr
̈
om (RN) [29], Bardeen [30, 42], Hay-
ward [31, 58], Kazakov-Solodhukin (KS) [33], and also the
asymptotically-flat Einstein-Maxwell-dilaton (EMd-1) with
3
While the electromagnetic and scalar charge parameters for the RN and
JNW spacetimes are allowed to take values
̄
q >
1
and
0
<
ˆ
̄
ν
:= 1
−
̄
ν <
1
respectively, they do not cast shadows for
̄
q >
√
9
/
8
and
0
.
5
≤
ˆ
̄
ν <
1
(see, e.g., Sec. IV D of [15] and references therein).
9
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
normalized physical charge
3
4
5
6
7
̃
r
sh
[
M
]
excluded region
allowed region
Schwarzschild
RN
RN
∗
Bardeen
Hayward
Frolov (
̄
l
= 0
.
4)
JNW
∗
KS
EMd-1
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
a
3
4
5
6
7
r
sh
,A
(
i
=
π/
2) [
M
]
excluded region
allowed region
Kerr
KN ( ̄
q
= 0
.
25)
KN ( ̄
q
= 0
.
95)
Bardeen ( ̄
q
m
= 0
.
25)
Bardeen ( ̄
q
m
= 0
.
75)
Hayward (
̄
l
= 0
.
25)
Hayward (
̄
l
= 0
.
75)
Sen ( ̄
q
m
= 0
.
25)
Sen ( ̄
q
m
= 1
.
25)
FIG. 2.
Left:
Shadow radii
̃
r
sh
for various spherically symmetric black-hole solutions, as well as for the JNW and RN naked singularities
(marked with an asterisk), as a function of the physical charge normalized to its maximum value. The gray/red shaded regions refer to the
areas that are
1
-
σ
consistent/inconsistent with the 2017 EHT observations and highlight that the latter set constraints on the physical charges
(see also Fig. 3 for the EMd-2 black hole).
Right:
Shadow areal radii
r
sh
,A
as a function of the dimensionless spin
a
for four families of
black-hole solutions when viewed on the equatorial plane (
i
=
π/
2
). Also in this case, the observations restrict the ranges of the physical
charges of the Kerr-Newman and the Sen black holes (see also Fig. 3).
φ
∞
= 0
and
α
1
= 1
[34, 35, 56] solution (see Sec. IV
of [15] for further details on these solutions). For each of
these solutions we vary the corresponding charge (in units of
M
) in the allowed range, i.e., RN:
0
<
̄
q
≤
1
; Bardeen:
0
<
̄
q
m
≤
√
16
/
27
; Hayward:
0
<
̄
l
≤
√
16
/
27
; Frolov:
0
<
̄
l
≤
√
16
/
27
,
0
<
̄
q
≤
1
; KS:
0
<
̄
l
; EMd-1:
0
<
̄
q <
√
2
, but report the normalised value in the figure
so that all curves are in a range between 0 and 1. The figure
shows the variation in the shadow size of KS black holes over
the parameter range
0
<
̄
l <
√
2
. Note that the shadow radii
tend to become smaller with increasing physical charge, but
also that this is not universal behaviour, since the KS black
holes have increasing shadow radii (the singularity is smeared
out on a surface for this solution, which increases in size with
increasing
̄
l
).
Overall, it is apparent that the regular Bardeen, Hay-
ward, and Frolov black-hole solutions are compatible with the
present constraints. At the same time, the Reissner-Nordstr
̈
om
and Einstein-Maxwell-dilaton 1 black-hole solutions, for cer-
tain values of the physical charge, produce shadow radii that
lie outside the
1
-
σ
region allowed by the 2017 EHT obser-
vations, and we find that these solutions are now constrained
to take values in,
0
<
̄
q
.
0
.
90
and
0
<
̄
q
.
0
.
95
respec-
tively. Furthermore, the Reissner-Nordstr
̈
om naked singular-
ity is entirely eliminated as a viable model for M87* and the
Janis-Newman-Winicour naked singularity parameter space is
restricted further by this measurement to
0
<
ˆ
̄
ν
.
0
.
47
. Fi-
nally, we also find that the KS black hole is also restricted to
have charges in the range
̄
l <
1
.
53
. In addition, note that
the nonrotating Einstein-Maxwell-dilaton 2 (EMd-2) solution
[37] – which depends on two independent charges – can also
produce shadow radii that are incompatible with the EHT ob-
servations; we will discuss this further below. The two EMd
0
.
00
0
.
25
0
.
50
0
.
75
1
.
00
1
.
25
̄
q
e
EMd-2
excluded
region
allowed region
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
1
.
2
1
.
4
̄
q
m
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
a
Sen
excluded
region
allowed region
FIG. 3. Constraints set by the 2017 EHT observations on the non-
rotating Einstein-Maxwell-dilaton 2 and on the rotating Sen black
holes. Also in this case, the gray/red shaded regions refer to the
areas that are
1
-
σ
consistent/inconsistent with the 2017 EHT obser-
vations).
black-hole solutions (1 and 2) correspond to fundamentally
different field contents, as discussed in [37].
We report in the right panel of Fig. 2 the shadow areal ra-
dius
r
sh
,A
for a number of stationary black holes, such as Kerr
[39], Kerr-Newman (KN) [40], Sen [41], and the rotating ver-
sions of the Bardeen and Hayward black holes [42]. The data
refers to an observer inclination angle of
i
=
π/
2
, and we
find that the variation in the shadow size with spin at higher
inclinations (of up to
i
=
π/
100
) is at most about
7
.
1%
(for
i
=
π/
2
, this is
5%
); of course, at zero-spin the shadow size