The Milky Way’s Supermassive Black Hole: How Good a Case Is It?

Abstract

The compact and, with ${\sim }4.3\pm 0.3\times 10^6$ M$_{\odot }$, very massive object located at the center of the Milky Way is currently the very best candidate for a supermassive black hole (SMBH) in our immediate vicinity. The strongest evidence for this is provided by measurements of stellar orbits, variable X-ray emission, and strongly variable polarized near-infrared emission from the location of the radio source Sagittarius A* (SgrA*) in the middle of the central stellar cluster. Simultaneous near-infrared and X-ray observations of SgrA* have revealed insights into the emission mechanisms responsible for the powerful near-infrared and X-ray flares from within a few tens to one hundred Schwarzschild radii of such a putative SMBH. If SgrA* is indeed a SMBH it will, in projection onto the sky, have the largest event horizon and will certainly be the first and most important target for very long baseline interferometry observations currently being prepared by the event horizon telescope (EHT). These observations in combination with the infrared interferometry experiment GRAVITY at the very large telescope interferometer and other experiments across the electromagnetic spectrum might yield proof for the presence of a black hole at the center of the Milky Way. The large body of evidence continues to discriminate the identification of SgrA* as a SMBH from alternative possibilities. It is, however, unclear when the ever mounting evidence for SgrA* being associated with a SMBH will suffice as a convincing proof. Additional compelling evidence may come from future gravitational wave observatories. This manuscript reviews the observational facts, theoretical grounds and conceptual aspects for the case of SgrA* being a black hole. We treat theory and observations in the framework of the philosophical discussions about “(anti)realism and underdetermination”, as this line of arguments allows us to describe the situation in observational astrophysics with respect to supermassive black holes. Questions concerning the existence of supermassive black holes and in particular SgrA* are discussed using causation as an indispensable element. We show that the results of our investigation are convincingly mapped out by this combination of concepts.

Appendices

Appendix 1: An Example for Establishing Existence Claims

As an example for establishing existence claims in the (recent) history of physics, we look at the case of atoms and molecules. We consider this case—because it has been widely discussed in philosophy of science—is the final acceptance of the existence of molecules/atoms due to the work of Jean Perrin on Brownian motion in the early 20th century as the collision with the quick atoms or molecules in the gas or liquid. Brownian motion itself was first discovered by the Scottish botanist Robert Brown in 1827 as random motion of particles.

Perrin received the Nobel Prize for physics in 1926 for his work that “put a definite end to the long struggle regarding the real existence of molecules” (C.W. Oseen, member of the Nobel Committee for Physics). So, how was Perrin able to establish the existence of molecules/atoms and what can we learn from this for the question of the existence of black holes? Atoms and molecules among other places played a major role in the kinetic theory of heat. However, there were debates about whether atoms/molecules were merely a useful fiction that yielded (in many cases) the right result or whether they should be accepted as real. For some time it appeared to be the case that this question could not be resolved by experimental means. In the early 20th century (partly through the work of Einstein) atoms and molecules became accessible, i.e. new theoretical means were devised so as to bring the hypothesis into the reach of established experimental methods (see [247], “Making Contact with Molecules: On Perrin and Achinstein”).

In making the question of atoms/molecules not only experimentally accessible but to decide it positively, three features proved to be important:

• i. The assumption that atoms/molecules exist puts constraints on observational data: The assumptions could only be true if some observable magnitudes had certain very definite values (Avogadro’s number/constant). This appears to be an essential point in a number of cases. What is asked for here are novel predictions, i.e. the prediction of phenomena, which would be very unlikely if atoms/molecules did not exist (compare the prediction of the deflection of light by the sun as evidence for GR; Karl Popper tried to capture this by his notion of a high degree of falsifiability). The prediction has to be risky.

• ii. On rival accounts, i.e. on the denial of atoms/molecules, the value of Avogadro’s number would have been completely different. Hence, the value of Avogadro’s constant is dependent on the existence of atoms.

• iii. The relevant value was experimentally established in a number of different/independent ways. Avogadro’s constant is tested in several different ways and today there are about 60 different and independent ways to measure it [25], making use of e.g. electrical charge and current, in determinating the properties of gas, liquids and crystals.

Following the ground breaking work of Perrin, today the acceptance of atoms is supported by direct imaging of molecules, atoms or even their bounds (e.g. [154, 228]).²⁷

Appendix 2: From (Anti)realism and Underdetermination to a Causal Principle

How for theory and observations the concepts of “Realism” and “Underdetermination” may be linked to causation is outlined in the following:

Theory Taken Literally as Being True Black holes are extremely simple objects and hence, ideally suited to be regarded as a kind of theoretical concept that historically has the tendency to be taken literally as real by realists (see below and statements by Psillos [246] and Sellars [275]). The problem is that other theories could be regarded as being similarly simple (although they are often more complex as they require the definition of additional quantities like particle masses and interaction forces etc.). The fact that one cannot easily distinguish between them experimentally, leaves enough freedom to being unable to decide if some other entities implied by a literal reading of the theory, e.g. a compact object like a black hole and its possible alternatives, are indeed real (e.g. the shadow of such a compact object, i.e. the presence, depth and size of an emission free or sparse region close to the position of the compact object; see Sect. 4.10).

Here, also the possibility of an experimental comparison between theory and implied entities (if interpreted as predicted phenomena like speeds of stars or emitting blobs, timescales associated with consecutive events, or structures like the shadow) in a way relieves many philosophers (especially if facts or statements are being taken literally) from the “commitment to a host of entities with a (to say the least) questionable ontic status: numbers, geometrical points, theoretical ideals, models and suchlike” [246].

Therefore, we must clarify that at this point we are not concerned with the question of whether there is reality to the theoretical concept of a black hole as such, for instance as a concept within the theory or relativity. The goal of the observational and experimental astrophysics is to test the properties of an observable entity against a theoretical concept in order to identify it in a commonly agreeable way with the object described by this very theoretical concept.

Reality of Entities Derived from Theory The entities implied by the theory of black holes (here taken as observational results or as implied physical scenarii, i.e. how stars orbit the black hole or—in the near future—how exactly the black hole shadow looks like) can also be regarded as real, as they all come (are derived) indeed “from theory and $\ldots$ there is no theory-free standpoint from which” [246], these entities can be viewed. This means, as valuable observational predictions are usually all derived from theory there is no theory-independent criterion of reality that can be used in an accepted way. The goal is to define and successfully observe entities i.e. predictions, for which a black hole plays an indispensable role in their explanation.

While this is clearly the preferred situation for the physicist (i.e. a successful comparison between observational results or the properties of implied physical scenarii) this is also true for theory as such from the viewing angle of realists that historically tend to read (take) theories literally as real. As this can be explained out of theory it can be regarded as a permissive criterion [275] that particularly does not disallow abstract entities from being real. Psillos [246] points out that “this explanatory criterion should not be confused with a causal criterion”, e.g. in the form of the Eleatic Principle (see below), “according to which being causally active, that is having causal powers, is a criterion of objecthood. Causal efficacy may well be a mark of reality, but not everything that is real is causally efficacious” [246]. The former would be good for experimentalists, the latter would be supported by black hole theory. Here, Psillos points at entities that are causally idle but causally relevant. Hence, he describes Colyvan’s “rounded out” version of the Eleatic Principle (Colyvan [54] and the following Appendix 3).

Causation as a Clue to Probe Existence For observed entities, may they be derived from theory or not, causation may be taken as a clue to make their existence plausible beyond doubt. David M. Amstrong states in his reply to Reinhardt Grossmann in the correspondence on the “Ontology and the Physical Universe” [57] that if one does not “postulate an entity [...] which is not required in ones account of causation”, then indeed causation can be used as a partial clue to the existence. The word “partial” is used here as it needs to be seen how the analysis of the causation needs to be done in every particular case. Goal of the observational astrophysics is to optimize this aspect.

As Andrew Newman states: “Talking about existing and talking about being real are just different ways of talking about the same thing” [222]. As supporting examples he quotes Gottlob Frege and Bertrand Russell who are guided to things that are real by syntax (e.g. in [45]), Quine, who demands that something must be quantifiable if it is called real (e.g. in [61]), and D.M. Amstrong demanding causal significance for real things (e.g. in [222]). Causal significance or relevance is an essence of any form of the Eleatic Principle (see below).

Andrew Newman points out [222]: “Alex Oliver claims that the Eleatic Principle is ambiguous because there is an epistemological reading of ‘reason’ and a metaphysical reading of ‘reason’, mostly worried about the existence of causally inactive entities.” This criticism might appear to be applicable to aspects of the black hole physics, however, all experiments aim at consolidating causal activity.

Reality and the Eleatic Principle In the case of black holes, we may make use of the possible claim that the physical world is causally closed, i.e. that all genuine causes of physical events are physical causes (e.g. [203]). Hence, in the case of black holes we do not run into the problem that this particular science entity is not reducible to a physical entity. Even if the reduction to such an entity is currently not fully conducted, all instrumental and observational efforts aim at improving the performance of that reduction.

Appendix 3: The Eleatic Principle

If underdetermination can be fought or even partially overcome, then causation may be used to further underline the realism or existence of an entity in a generally acceptable way. This involves the usage of a causal criterion that may be in the form of the Eleatic Principle (for a general overview see e.g. [54, 55]). Colyvan [54] gives a consice definition of the classical Eleatic Principle “An entity is to be counted as real if and only if it is capable of participating in causal processes”.

The principle is named after a Greek school in lower Italy Elea (’$E \lambda \acute{\epsilon } \alpha$) closely linked to the philosophers Parmenides, Zeno and Xenophanes of Colophon (going back to a figure in Plato’s dialogue Sophistes). As a philosophical conceptual aspect, the School of Elea rejects any epistemological criteria simply based on sensual experiences (Fig. 2). According to the Eleatic Principle, we should be realists about whatever manifests itself in virtue of having effects (in ‘A World of States of Affairs’ by Amstrong [11], see also the discussion in “Quining Naturalism” by Price [241]). The use of the Eleatic Principle in our context needs some clarification. The classical version of the Eleatic Principle uses and connects (in a more time-like fashion) mainly causally active entities. In this version non-causal entities like abstract mathematical and physical concepts are regarded as causally idle and therefore they are not considered as suitable entities. This problem has been explained in detail by Colyvan in his article “Can the Eleatic Principle be Justified?” [54]. He investigates several justification attempts and shows how and when they fail or what their drawbacks are. In order to avoid these problems he suggests to use a “rounded out” version of the Eleatic Principle in which also causally idle entities can be used. This “rounded out” version contains the classical version of the Eleatic Principle (Fig. 2). In the case of the Galactic Center and supermassive black holes in general, involvement of at least the “rounded out” version of the Eleatic Principle is justified (see below and [54]).

Mark Colyvan’s version of the “rounded out” Eleatic Principle does not stand alone. He points out that this more general principle that he proposes has great similarities to Quine’s thesis that “we are only ontologically committed to all and only the entities that are indispensable to our current best scientific theories” (see [249]). Indispensable entities then include both causally idle and causally efficacious entities. A critical discussion of the Eleatic Principle with a focus on the role of mathematical objects and Colyvan’s defense of Quine’s indispensability argument is given by Marcus [185]. It should also be mentioned that the entities we connect in the synthesis Sect. 7 have by themselves a rather complex nature. In most cases they are the result of a logical combination of causally idle and active entities and have a structure similar to that shown in Eq. (9). Causally idle (but ally relevant) entities occur e.g. through the theory of image formation in all wavelength domains or in relativistic concepts that are used to extract expected situations and compare them to measurements. In particular the “data inversion” problem in the image formation theory of interferometry might well serve as an example of true “underdetermination” in astrophysics. Causally efficacious entities occur e.g. in the form of stars, pulsars, emitted radiation or astrophysical instruments and telescopes, all of which can be located in spacetime in contrast to the causally idle entities.

Appendix 4: Time Scales and Luminosities of Gravitational Wave “Ringing”

The analysis is done based on the formalism presented in Misner et al. [202] and Shapiro and Teukolsky [279]. The tidal disruption radius $r_\mathrm{{T}}$ for main-sequence stars with mass $m_{\star }$ and radius $R_{\star }$ is larger than the Schwarzschild radius $R_\mathrm{{S}}$ of the supermassive black hole,

(20)

so we cannot expect any gravitational wave event from main-sequence star inspirals, since they will be tidally disrupted. On the other hand, Schwarzschild radius is larger than tidal disruption radius for white dwarfs with $R_{\star }=0.01\,R_{\odot }$, neutron stars with $R_{\star }=10^{-5}\,R_{\odot }$, and stellar black holes, so these are not subject to tidal disruption.

A circularized orbit of a compact star with the semimajor axis $a_{\star }$ around the supermassive black hole emits gravitational waves at a frequency

(21)

reaching $\sim 1.1\,\mathrm {mHz}$ at the innermost stable circular orbit.

The characteristic scaling amplitude for the EMRI event at the distance of the Galactic center is

(22)

and the effective metric perturbation is given by the amplitude $h_0$ times the square root of the number of cycles the EMRI spends in the band with a bandwidth $\Delta \nu$ centered at the frequency $\nu$. The number of cycles N is given by $N=\nu ^2/\dot{\nu }=\nu \tau$, where the characteristic evolution time-scale $\tau \equiv \nu /\dot{\nu } = 8/3(t_\mathrm{coal}-t) \propto \mathcal {M}^{-5/3} \nu ^{-8/3}$ and $t_\mathrm{coal}$ is the coalescence time. The ‘chirp mass’ may be expressed as , where the approximation holds for the EMRI. Finally, the metric perturbation or basically the signal is proportional to .

The expected luminosity $L_\mathrm{GW}$ averaged over one period and evaluated at the innermost stable circular orbit [233] is

(23)

which is valid for circular orbits. In general, the equation 23 depends strongly on the orbital eccentricity [233]. The incoming flux $F_\mathrm{GW}$ may be expressed as a function of the amplitude $h_0$ and the observed frequency $\nu _\mathrm{{GW}}$,

$$\begin{aligned} F_\mathrm{GW}\simeq 0.005 \left( \frac{\nu _\mathrm{GW}}{1\,\mathrm{mHz}}\right) ^2 \left( \frac{h_{0}}{4\times 10^{-18}\,} \right) ^2\,\mathrm{erg\,cm^{-2}\,s^{-1}}\,. \end{aligned}$$

(24)

The coalescence (merger) time-scale $\tau _\mathrm{{merge}}$ depends on the eccentricity of the orbit e. For a circular orbit,

(25)

where $a_0$ is an initial semi-major axis. For initially highly eccentric orbits, $e\approx 0.99$, it reduces to $\tau _\mathrm{{merge}}=768/425 \tau _0(a_0)(1-e_0)^{7/2}\approx 0.016\,\mathrm {year}$.

The detection of an EMRI event associated with the Galactic center black hole would enable us to precisely measure the mass and the spin of the black hole with an independent measurement. The waveform could also reveal deviations from the Kerr metric as well as a possibly different character of a central compact object (e.g. a boson star). An exemplary waveform for an EMRI event for the ratio and the black hole spin $J=0.998$ is depicted in Fig. 9. The inspiral starts at four gravitational radii.

The EMRI events from the Galactic center will be within the detection sensitivity limits of the planned LISA and eLISA space-base interferometers that will be capable to detect the gravitational wave events with low frequencies in the range $\nu _\mathrm{{GW}}=0.1$–$1\,\mathrm {mHz}$. Although the likelihood to detect an EMRI event for the Galactic center is rather low during the mission lifetime, [117] estimate that LISA will be able to detect ${\sim }2$ EMRI events of $1.4\,M_{\odot }$ compact objects (white dwarfs and neutron stars) per cubic Gpc per year for black holes with similar masses as Sgr A*.