The determination of incidence and prevalence of secondary treatment failure (STF) is even more complex. So far, there is no clear definition of STF. If no response to a BoNT injection can be detected neither by the patient nor the treating physician, the diagnosis of a complete treatment failure (cSTF) is comparably easy [16]. However, this is the end stage of a longer process, starting with an increasing reduction of the duration of action of a BoNT injection before the 4-week peak effect becomes reduced. This has been described in 2004 by Dressler in detail [1].

When patients are reinjected every 3 months, the reduction of the duration of action cannot be measured directly but results in a systematic worsening of disease severity (for details see [5, 17]). We, therefore, have proposed a formal definition of pSTF in patients with CD. If a patient has responded with more than three points on the TSUI scale [18] and then develops a systematic worsening over three injection

cycles of more than two TSUI score points and reports a reduction of the effect of BoNT injection to head position, tremor, or pain, a pSTF has to be suspected.

Our experience is that such a formal definition of pSTF is helpful to detect patients with pSTF. Furthermore, in a cohort of 32 patients with pSTF according to this definition, 25 patients (= 78%) had a positive MHDA test [4]. This is definitely more than in a large cohort of patients with suspected pSTF without formal definition were only about 50% were MHDA-positive [19].

This led to the opinion that NAB-associated secondary nonresponse (SnR) is a rare occurrence, and SnR is more frequently due to an insufficient dose, inappropriate muscle selection, or improper injection technique or targeting [10, 20]. But to our opinion, these aspects of BoNT treatment do not suggest the development of pSTF but indicate insufficient and inappropriate BoNT treatment. pSTF can only be suspected when the patient worsens despite of therapy optimization.

In a cross-sectional study on 66 patients with CD who had started their BoNT therapy with abo- or onaBoNT/A, we have analyzed patients´ drawing of the course of disease (course of disease graphs (CoDGs)) after the onset of BoNT therapy over the entire duration of BoNT therapy. Five different response types could be distinguished: the rapid or golden responder (RR) type, the continuous response (CR) type, the poor response (PR) type, and the secondary treatment failure (STF) type I and II.

The RR type is characterized by a rapid response after the onset of BoNT therapy, followed by a further less rapid improvement. The CR type is characterized by a continuous improvement over the entire duration of treatment, The PR type is characterized by an improvement of less than 20%. Patients who drew a PR type CoD graph were primary nonresponders. The STF types I and II are characterized by an initial improvement followed by a secondary worsening. In the STF type II, the second period of improvement followed after the switch of the BoNT preparation.

#### **Figure 2.**

*The severity of cervical dystonia versus percentage of time elapsed from treatment onset. A. Mean course of disease graph (CoD graph; solid line) plus 1 standard deviation range (hatched lines) of 11 CD patients who had drawn a STF type I CoD graph after the onset of BoNT therapy. The severity of CD rapidly decreases initially but then worsens again. B. Mean course of disease graph (CoD graph; solid line) plus 1 standard deviation range (hatched lines) of six CD patients who had drawn a STF type II CoD graph after the onset of BoNT therapy. The severity of CD improved initially, then worsened again, but improved a second time after the switch of the BoNT preparation.*

*Clinical Relevance of Neutralizing Antibodies in Botulinum Neurotoxin Type A DOI: http://dx.doi.org/10.5772/intechopen.102896*

Among the 66 patients, 17 patients (= 25.8%) produced a STF type. In **Figure 2A**, the mean CoD graph (full line) plus/minus 1 standard deviation (hatched lines of the 11 patients with a STF type I CoD graph is presented. In **Figure 2B**, the corresponding mean CoD graph (full line) plus/minus 1 standard deviation of the six STF-type II CoD graphs are presented.

The same task (to draw the course of disease after the onset of BoNT therapy) was analyzed in 34 patients who had exclusively been treated with incoBoNT/A. No patient in the incoBoNT/A monotherapy group produced a STF type I or II CoD graph. This difference in frequency of drawing STF type I or II graphs is significant (*p* < .05).

In summary, partial secondary treatment failure (pSTF) occurs more frequently than positive NAB tests suggest. This indicates that NAB tests are less sensitive to detect pSTF than careful clinical investigation and patient's assessment of the efficacy of BoNT therapy.

#### **3. Implications of the presence of NABs on long-term outcome**

As mentioned, earlier, the presence of NABs does not imply that there is no clinical response at all. We have recently described a small cohort of nine CD patients with positive NAB testing in 2010 who did not want to be switched to another BoNT preparation. They were retested in 2013 and 2017. Their relation between TSUI scores and paralysis times in 2010, 2013, and 2017 are presented in **Figure 3A** (open symbols indicate data from 2010, full symbols indicate data from 2017, and missing symbols indicate data from 2013). Apart from two exceptional cases, little changes in paralysis times and TSUI scores can be observed.

#### **Figure 3.**

*Time to paralysis versus TSUI. A. Temporal development of the relation between TSUI scores and paralysis times in 2010, 2013, and 2017 in nine CD patients in whom the complex protein-containing preparation had not been switched, although the initial MHDA test was positive (paralysis time > 60 mins). Open circles indicate values of the investigation in 2010, full circles indicate values of the investigation in 2017, and no circles indicate values of the investigation in 2013. B. Temporal development of the relation between TSUI scores and paralysis times in 2010, 2013, and 2017 in nine CD patients in whom the complex protein-containing preparation was switched to incoBoNT/A in 2010. Open squares indicate values of the investigation in 2010, full squares indicate values of the investigation in 2017, and no squares indicate values of the investigation in 2013. Apart from one exceptional case, paralysis times of eight patients decrease between 2010 and 2017. In six out of nine patients, TSUI scores improve, but the improvement of the TSUI score is less pronounced compared to the improvement of the paralysis times.*

The data of these nine CD patients in whom BoNT/A preparation had not been switched between 2010 and 2017 were compared to data of nine CD patients in whom the complex protein-containing BoNT/A preparation was switched to the complex protein-free incoBoNT/A preparation in 2010. The relation between TSUI scores and paralysis times of these nine switchers is presented in **Figure 3B**. Apart from one exceptional case, paralysis times improved in the switchers. In parallel, the TSUI scores also improved in six out of nine patients, but the improvement of TSUI scores was less pronounced than the improvement of the paralysis times (see **Figure 3B**).

In summary, so far little is known about the development of NAB titers when patients remain on their BoNT preparation under which they have developed NABs. However, there is increasing evidence that switching from a complex protein-containing preparation (abo- and onaBoNT) to a complex protein-free preparation (incoBoNT/A) may lead to a significant long-lasting improvement of paralysis times [4, 21].

#### **4. No NAB induction in patients under incoBoNT long-term monotherapy**

Soon after the treatment of patients with the old "Botox," it became obvious that in a large percentage of patients, NABs were induced [22]. This led to a purification process of the "old" Botox® preparation, a reduction of the protein load of the "new" Botox® by a factor of 5–6 [10, 23], and an improvement of the antigenicity of the Botox® preparation [24]. The incidence of NAB induction reported for the "new" Botox® preparation was around 1%/year [24].

In 2005, incoBoNT was licensed for the treatment of CD [25]. By removal of botulinum neurotoxin complex proteins and elimination of biological inactive fragments of the BoNT molecule, the protein load of this preparation was reduced down to 0.55 ng compared to 4.8 ng of the Botox and 5.3 ng of the Dysport® preparation [3, 26]. But so far, no convincing study has been presented that the lower protein load of the incoBoNT preparation also leads to a significantly lower antigenicity compared to the other two BoNT/A preparations licensed in Europe.

However, it has not been reported that NABs or pSTF were induced in a patient who had exclusively been treated with incoBoNT/A [15]. Cases with NABs have been presented who had been treated only for a few treatment cycles with abo- or onaBoNT and then were switched to incoBoNT/A [1, 15], but NAB induction under incoBoNT/A monotherapy has not been observed.

This is in line with a recent observation that in 34 CD patients who were exclusively been treated with incoBoNT/A, no patient had a positive MHDA test [15] and no patient drew a STF type I or II CoD graph (see Section 2 above).

In summary, the antigenicity of the complex protein-free incoBoNT/A preparation appears to be very low, since no NAB induction has been observed in patients who have exclusively been treated with incoBoNT/A.

#### **5. Prediction of the NAB presence from clinical treatment-related data**

It has been demonstrated that long-term treated CD patients with NABs have a significantly worse TSUI score, were treated with significantly higher doses, and have higher pain scores of the CDQ24 than long-term treated CD patients with a negative MHDA [12, 27, 28]. The paralysis times of the MHDA are significantly correlated with the actual doses, the actual TSUI scores, and the CDQ24 pain scores [28].

*Clinical Relevance of Neutralizing Antibodies in Botulinum Neurotoxin Type A DOI: http://dx.doi.org/10.5772/intechopen.102896*

To analyze which treatment-related parameter may be used to predict the presence of NABs best, NABs were determined in 59 patients with pSTF. Patients 'assessments of the effect of long-term BoNT/A treatment were determined by asking patients to estimate the improvement of CD in percent of the initial severity of CD at the onset of BoNT therapy (IMPQ) and to mark the actual severity of CD in percent of the initial severity on a visual analog scale which yielded a second estimation of improvement (IMPD). The receiver operating characteristics (ROC) curves for the prediction of the presence of NABs by IMPQ and IMPD are presented in **Figure 4A**. The sensitivity and specificity of both parameters were around 0.7–0.8.

The treating physicians scored the actual severity of CD by means of the TSUI score (ATSUI) and determined the improvement since the onset of BoNT therapy by calculating the difference between ATSUI and the initial severity of CD at the onset of BoNT therapy (IMPTSUI) in the same 59 patients. Similar ROC curves were calculated for ATSUI and IMPTSUI (**Figure 4B**). Treating physicians´ scoring predicted the presence of antibodies less well compared to the assessment of the patients. Sensitivity was lower than 0.6, and specificity was also between 0.7 and 0.8.

In summary, patients realize the NAB-induced reduction of the efficacy of BoNT injections quite well, probably better than treating physicians scoring of the treatment effect.

#### **Figure 4.**

*The ROC curves for prediction of the presence of NABs by IMPQ and IMPD. A. ROC curves analyzing the relation between the presence of NABs and patients´ assessments of the improvement of CD since the onset of BoNT therapy: Solid circles indicate the ROC curve of IMPQ and light circles indicate the ROC curve of IMPD. For both parameters, sensitivity and specificity lie around 0.7–0.8. B. ROC curves analyzing the relation between the presence of NABs and treating physicians' scoring of the actual severity of CD (ATSUI) and the improvement of CD since the onset of BoNT therapy (IMPTSUI): Solid squares indicate the ROC curve of ATSUI and light circles indicate the ROC curve of IMPTSUI. For both parameters, the sensitivity is around 0.5, whereas the specificity lies around 0.7–0.8.*

#### **6. Conclusion**

Induction of NABs occurs frequently (**Figure 1**), may become manifest after years of successful treatment (**Figure 3**), progresses with the duration of treatment (**Figure 1**), and has clinical implications. In patients with CD, it goes along with higher severity of

CD, leads to more pain, and affords treatment with increasingly higher doses. Patients realize the reduction of efficacy of BoNT/A treatment quite well (**Figure 4**).

Since the induction of NABs has not been observed under monotherapy with incoBoNT/A and switch to incoBoNT/A may lead to clinical improvement in patients with pSTF after ona- and aboBoNT/A incoBoNT/A seems to have a very low antigenicity. We, therefore, recommend using the complex protein-free BoNT/A preparation incoBoNT/A from the very beginning of BoNT/A therapy to reduce the risk of antibody formation as low as possible.

### **Acknowledgements**

We would like to thank the following individuals for their expertise and assistance throughout all aspects of our studies especially Prof. Dr. Philipp Albrecht for making substantial conceptual and design contributions, Dr. Marek Moll, Beyza Ürer, Raphaela Brauns, and Dietmar Rosenthal for contributing to gathering data conducting the data analysis and creating the tables and figures.

### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

Harald Hefter and Sara Samadzadeh\* Department of Neurology, University of Düsseldorf, Düsseldorf, Germany

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Clinical Relevance of Neutralizing Antibodies in Botulinum Neurotoxin Type A DOI: http://dx.doi.org/10.5772/intechopen.102896*

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[3] Frevert J. Content of botulinum neurotoxin in Botox®/Vistabel®, Dysport®/Azzalure®, and Xeomin®/ Bocouture®. Drugs R & D. 2010; **10**(2):67-73

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[6] Oshima M, Deitiker PR, Jankovic J, Duane DD, Aoki KR, Atassi MZ. Human T-cell responses to botulinum neurotoxin. Responses in vitro of lymphocytes from patients with cervical dystonia and/ or other movement disorders treated with BoNT/A or BoNT/B. Journal of Neuroimmunology. 2011;**240-241**:121-128

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[8] Dolimbek BZ, Aoki KR, Steward LE, Jankovic J, Atassi MZ. Mapping of the regions on the heavy chain of botulinum neurotoxin A (BoNT/A) recognized by antibodies of cervical dystonia patients with immunoresistance to BoNT/A. Molecular Immunology. 2007;**44**(5):1029-1041

[9] Atassi MZ, Dolimbek BZ, Jankovic J, Steward LE, Aoki KR. Regions of botulinum neurotoxin A light chain recognized by human anti-toxin antibodies from cervical dystonia patients immunoresistant to toxin treatment. The antigenic structure of the active toxin recognized by human antibodies. Immunobiology. 2011;**216**(7):782-792

[10] Bellows S, Jankovic J. Immunogenicity associated with botulinum toxin treatment. Toxins (Basel). 2019;**11**(9):491

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[12] Hefter H, Rosenthal D, Moll M. High botulinum toxin-neutralizing antibody prevalence under long-term cervical dystonia treatment. Movement Disorders Clinical Practice. 2016;**3**(5):500-506

[13] Albrecht P, Jansen A, Lee JI, Moll M, Ringelstein M, Rosenthal D, et al. High prevalence of neutralizing antibodies after long-term botulinum neurotoxin therapy. Neurology. 2019;**92**(1):e48-e54

[14] Contarino MF, Van Den Dool J, Balash Y, Bhatia K, Giladi N, Koelman JH, et al. Clinical practice: Evidence-based recommendations for the treatment of cervical dystonia with botulinum toxin. Frontiers in Neurology. 2017;**8**:35

[15] Hefter H, Brauns R, Ürer B, Rosenthal D, Albrecht P. Effective longterm treatment with incobotulinumtoxin (Xeomin®) without neutralizing antibody induction: A monocentric, cross-sectional study. Journal of Neurology. 2020;**267**(5):1340-1347

[16] Walter U, Mühlenhoff C, Benecke R, Dressler D, Mix E, Alt J, et al. Frequency and risk factors of antibody-induced secondary failure of botulinum neurotoxin therapy. Neurology. 2020;**94**(20):e2109-e2e20

[17] Hefter H, Spiess C, Rosenthal D. Very early reduction in efficacy of botulinum toxin therapy for cervical dystonia in patients with subsequent secondary treatment failure: A retrospective analysis. Journal of Neural Transmission (Vienna). 2014;**121**(5):513-519

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[20] Bellows S, Jankovic J. Reply to comment on re-visiting immunogenicity associated with botulinum toxin treatment. Toxins. 2019;**11**:491 Toxins. 2020;12(2):72

[21] Hefter H, Hartmann C, Kahlen U, Moll M, Bigalke H. Prospective analysis of neutralising antibody titres in secondary non-responders under continuous treatment with a

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[22] Greene P, Fahn S, Diamond B. Development of resistance to botulinum toxin type A in patients with torticollis. Movement Disorders. 1994;**9**(2):213-217

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### *Edited by Suna Sabuncuoglu*

*Botulinum Toxin - Recent Topics and Applications* provides doctors and researchers with the most up-to-date information on the structure and action of botulinum toxins as well as their usage to treat a wide range of disorders. The book covers the fundamentals of botulinum toxins, including improvements in our understanding of their molecular structure and mechanism of action. Furthermore, it covers the production and formulation of botulinum toxins for clinical applications. The information contained herein will hopefully shift the paradigm of botulinum toxin research, moving it away from a focus on toxin attributes toward a focus on toxin engineering based on patient requests and physician experiences in everyday practice.

Published in London, UK © 2022 IntechOpen © Anna Kim / iStock

Botulinum Toxin - Recent Topics and Applications

Botulinum Toxin

Recent Topics and Applications

*Edited by Suna Sabuncuoglu*


Neuroprotective and Neurorestorative Properties of Copolymer-1: Its Immunomodulating Effects on Ischemic Stroke

*María Yolanda Cruz Martínez, Melanie Tessa Saavedra Navarrete and José Juan Antonio Ibarra Arias*


**2. Grape and wine industry in China**

In the past decades, the area used for grape cultivation and the total wine production and consumption in China has rapidly expanded. Relevant statistics regarding the grape and wine industry since the birth of the People's Republic of China are shown in **Table 1**.

As can be seen from **Table 1** below, China has accomplished great success in the grape and wine industry with unprecedented speed, both in terms of vineyard area, wine production, and consumption. According to the latest International Organization of Vine and Wine (OIV) report on the world Viti vinicultural situation (2019 and 2020) [4, 5], the size of the total world area under vines (regardless of the final destination of the grapes and including vineyards not yet in production) remained stable at 7.3 mha (millions of hectares) in 2020. With 961 kha, Spain remains the clear leader in terms of cultivated vine area, followed by France (797 kha) and China (785kha).

The world wine production (excluding juice and musts) in 2020 was estimated at 258 mhl as Italy (49.10 mhl) maintained its position as the world's leading producer, followed by France (46.60 mhl) and Spain (40.70 mhl). China, on the other hand, produced 6.60 mhl. The data shows a slight drop in global wine consumption (estimated at around 234 mhl) in 2020 because of the COVID-19 outbreak. The United States (33.0 mhl), France (24.7mhl), and Italy (24.5 mhl) maintained their top three positions as the world's largest consuming countries with China ranking sixth with 12.4 mhl consumption in the world.

In China, Red varieties account for nearly 80% of the total vineyard area, while the white varieties proportion was only 20% [3]. Red wine is also far more popular *Winemaking in Cold Regions with Buried Viticulture in China DOI: http://dx.doi.org/10.5772/intechopen.99614*

*Note: It is estimated that wine grape production area only occupies 10% of the total vineyard. Units: kha, thousands of hectares; mt, millions of tons; khl, thousands of hectolitres; and mhl, millions of hectolitres.* **Source:** *National Bureau of Statistics in China (vineyard area and grape production), and China alcoholic drinks association (wine production).*

#### **Table 1.**

*The vineyard area, grape production, and wine production in China.*

in the Chinese market than other types of wine, and a large section of the population refers to such wine as "红酒" (Hóngjiu), because of its red color. ˇ


Preface

Chalcogenides have attracted the attention of the research community for decades because of their distinctive characteristics. In recent years, chalcogen-based materials have been used in various micro-electronics and energy devices, owing to their high feasibility, good mechanical properties, tunable electronic structures, and attractive electrical conductivity. This book describes the synthesis of chalcogen-based materials and their various uses. It is divided into three sections: "Chalcogen Fabrication", "Solar Cells and Batteries" and "Nanoscale Devices".

Section 1 includes four chapters that discuss the preparation of chalcogens for different applications. Chapter 1 describes the recent developments of multifunctional characteristics of chalcogen thin films. It also provides a detailed overview of the capability of chalcogen-based materials in supercapacitors, perovskite solar cells, thin-film sensors, and tandem solar cells. Chapter 2 designs the physical properties calculation of beryllium selenide (BeSe) by M.A. Ghebouli and B. Ghebouli. First-principles calculations were carried out using Cambridge Serial Total Energy Package (CASTEP) code. The authors estimate the hybrid functional parameters such as structural, elastic, electronics, and optical properties of zinc-blende and NiAs phase of BeSe. Chapter 3 summarizes spectroscopic studies of mechanically exfoliated PtSe2. It describes the electronic band structure and density of states for monolayer, bi-layer, tri-layer, and bulk PtSe2. Theoretical approximations determine the structural behavior of 1T phase PtSe2. Chapter 4 describes a series of synthetic organoselenium compounds such as ebselen analogues, diaryl selenides, spirodioxyselenurane, and spirodiazaselenuranes and their glutathione peroxidase (GPx) catalytic activity.

Section 2 includes two chapters on energy-based applications of chalcogenides such as perovskite solar cells and batteries. Chapter 5 describes the fabrication of perovskite solar cells using transition metal dichalcogenide (TMDs) as the electron transport layer. Moreover, it investigates the role of TiS2-, MoS2-, and WS2-based electron transport layers in perovskite solar cells. Chapter 6 discusses chalcogen (sulfur, selenium, and tellurium (SSTs))–tuned batteries. It describes aluminium ion batteries supported by SST cathodes. It also demonstrates various chalcogen substitutions in Li-ion and metal-ion batteries.

Section 3 includes chapters related to the fabrication of semiconducting nanodevices using chalcogen materials. Chapter 7 discusses the temperature-dependent assessment of conductivity, terahertz generation, and charge carrier dynamics. It also

discusses chalcogenides such as bismuth selenide, bismuth telluride, and antimony telluride and their charge movements. Finally, Chapter 8 describes the recording characteristics of chalcogenides.

This edited volume discusses the preparation of chalcogenides and their circular angle uses in different industries.

> **Dhanasekaran Vikraman** Division of Electronics and Electrical Engineering, Dongguk University-Seoul, Seoul, Korea

Section 1


**Black but Not Dark**

Andrzej Radosz, Andy T. Augousti and Pawel Gusin

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.77963

#### Abstract

Large black holes of millions of solar masses are known to be present in the centre of galaxies. Their mass is negligible compared to the mass of the luminous matter, but their entropy far exceeds the entropy of the latter by 10 orders of magnitude. Strong gravitational fields make them 'black'—but at the same time, they cause them to emit radiation so they are not 'dark'. What is the meaning of their borders that may only be crossed once and that leads to the information paradox and what are the properties of their interiors? In discussing these and related questions (is it possible that the volume of a black hole might be infinite?), we uncover the unexpected meaning of the term 'strong gravity'.

Keywords: gravity, black holes, horizon, interior, information paradox

### 1. Introduction

Black holes (BHs) are sources of the strongest gravitational fields in the Universe. On the other hand, they are also the outcomes of these strong gravitational fields. The first time they appeared in science was as a result of speculation. At the end of the XVIIIC, the English geologist (and astronomer) John Michell and the famous French mathematician Pierre-Simon Laplace independently considered the consequences of the presence of a large, compact massive object producing gravitational fields so strong that even light could not escape from them. For obvious reasons, discussions of this kind were limited in their nature at that time.

The next step came at the beginning of 1916, when Karl Schwarzschild, a mathematician and an army officer, found a specific solution for the field equations of Einstein's General Theory of Relativity. He found the solution for the particular case of a static, spherically symmetric spacetime. Schwarzschild sent the results of these studies to Albert Einstein in the form of

© 2018 The Author(s). Licensee IntechOpen. Distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License (https://creativecommons.org/licenses/by-nc/4.0/), which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited.

two chapters. The second of these two chapters contained what was, to Einstein, a controversial result. If the mass of the source of the gravitational field was both big enough and compact enough, then the solution was singular: a particular element of the metric tensor, a tool for describing the geometric properties of the spacetime, became infinite at some distance from the centre. Einstein was concerned by this effect and consequently had been slow to respond; in the meantime, Schwarzschild had died.

Schwarzschild's solution [1] (see subsequent text) reveals a specific form of behaviour and leads to the conclusion that in some circumstances, a so-called horizon (termed an event horizon) is formed around the black hole. Such a horizon acts as a semi-permeable 'membrane' [2]: it may be crossed only once and in one direction only. The radius of the event horizon is called the gravitational radius or the critical or Schwarzschild radius.

The term 'Black Hole', proposed in the 1960s by J.A. Wheeler, represents the reality of a strong gravitational field in which neither massive nor massless objects (i.e. light in the form of photons) could leave its interior. Black holes (BHs) had been regarded as hypothetical objects even as late as the early 1970s; at that time, a famous bet between two prominent physicists, Kip Thorn (Nobel Prize winner in Physics in 2017) and Stephen Hawking, was set. The subject of the bet was the experimental confirmation of the presence of black holes (the annual delivery of a journal from a building sector was the pledge for this bet).

Currently, it is assumed that there is a massive BH with a mass of millions of solar masses ðM⊙Þ in the centre of each large galaxy [3]. The black hole closest to the Solar System is located at a distance of 1700 light years from us. In the centre of Milky Way, there is a BH of mass 4:3∙106 M⊙; one of the largest BHs with a mass of a billion solar masses has been found in the centre of the Sombrero galaxy. This allows us to estimate that the matter confined within black holes is many orders of magnitude smaller than the luminous matter (LM) in each galaxy,

$$
\rho\_{\rm BH} \le 10^{-3} \rho\_{\rm LM} \tag{1}
$$

Hence contributes a negligible fraction of the total energy density. An interesting fact, however, is that the total entropy of black holes, SBHð Þ tot is 10 s of orders of magnitude higher than the entropy of radiation (CMB), estimated at a value of 1090. Indeed, the entropy of a BH of mass 4:3∙106 M⊙ is

$$S\_{BH} \left( 4.3 \cdot 10^6 M\_{\odot} \right) \cong 10^{90} \tag{2}$$

(see subsequent text), so

$$S\_{BH}(tot) \ge 10^{101} \,\text{,}\tag{3}$$

some 20 orders of magnitude smaller than the maximal entropy of our Universe.

The purpose of this exposition is to illuminate the properties of strong gravitational fields. This will be achieved via a discussion of particular processes and phenomena in the vicinity of the event horizon of black holes, on both sides of this horizon.

### 2. The Schwarzschild solution and the event horizon

x Let us consider the case of mass M as the source of a static and isotropic gravitational field. Then, the geometric properties of the resulting spacetime are determined by the Schwarzschild solution, a metric tensor ɡαβ. The line element, given in terms of Schwarzschild coordinates, α f g ¼ t,r, θ,φ, is (see [1])

$$ds^2 = g\_{a\beta}dx^a dx^\beta = f(r)c^2dt^2 - \frac{1}{f(r)}dr^2 - r^2d\Omega^2\tag{4}$$

rɡ where , rɡ ¼ 2GM f rð Þ¼ 1 � denotes the gravitational radius and dΩ2 ¼ dθ2 þ sin 2θdφ2 is a r c2 surface element of a unit sphere (we will utilize the system of units such that c ¼ G ¼ 1). Solution (2.1) is determined in an empty space outside mass M. Usually, when one deals with a weak gravitational field, the radius RM of mass M is much larger than its critical radius, RM ≫ rɡ, then f rð Þffi 1. Actually, for the Earth, rɡð Þ E ≈ 6 mm, the strength of the gravitational field is of the order of 10�9 ; the strength of the solar gravitational field is still very weak, 10�6 ; 1 but neutron stars yield strong gravitational fields, 10� . Black holes are the sources of the strongest fields, where an event horizon (defined by f rð Þ¼ 0) is developed. In such a case, we shall consider that the space outside and inside the horizon is empty—the mass of the black hole is confined at r ¼ 0—a singularity. This case will be referred to as an eternal black hole. We shall call the region outside the horizon as the exterior and that inside the horizon as the interior of the black hole.

### 3. Exterior of the Schwarzschild BH

The meaning of a strong gravitational field is revealed via investigation of the properties of the exterior and then the interior of a BH. It is natural to start from the former region. Let us underline the first, nearly trivial fact that the (relativistic) definition of the gravitational radius as the singularity of the metric (2.1) coincides with a purely classical physics definition of a critical radius such that the escape velocity becomes equal to the speed of light in a vacuum, c (see [4]). The generalization of this observation [4, 5] leads to the conclusion that the speed of a freely falling test particle tends to c, independently of the initial conditions. This and the other properties of the exterior of the event horizon may be described by means of geodesics of both kinds, that is, for massive and massless particles (light rays). The geodesic equations may be derived from the following Lagrangian (see Eq. (4)):

$$\mathcal{L} = f(r)\dot{t}^2 - \frac{1}{f(r)}\dot{r}^2 - r^2\dot{\theta}^2 - r^2\sin^2\theta\dot{\phi}^2\tag{5}$$

μ � dxμ in a standard manner leading to the Euler–Lagrange equations; x\_ dσ and σ is an auxiliary parameter. There are two conserved quantities resulting from the symmetry conditions: energy, e (due to time independence of the Lagrangian), and angular momentum, l (due to the invariance of the Lagrangian with φ). The latter condition results in the planar character of geodesic motions, so one may, without loss of generality, choose an equatorial plane, θ ¼ π and 2 express these conservation laws as follows:

$$f(r)\dot{t} = e,\tag{6}$$

$$r^2 \dot{\varphi} = l.\tag{7}$$

One can determine then arbitrary geodesics from the normalization condition

$$g\_{\mu\nu}\dot{\mathbf{x}}^{\mu}\dot{\mathbf{x}}^{\nu} = \eta \tag{8}$$

where η ¼ 1 or 0 for time-like (massive object) geodesics or for light-like (massless object) geodesics, respectively. Indeed, the radial component of the velocity vector, u (η ¼ 1), or the wave vector, k (η ¼ 0), takes the form:

$$
\dot{r} = \pm \sqrt{e^2 - f(r) \left(\frac{\partial}{\partial \tau} + \eta \right)} \tag{9}
$$

Using Eqs. (6)–(9), one can characterize both types of geodesics and illustrate in this way selected features of gravitational fields outside the BH horizon.

Apart from geodesic motions, we will also be employing systems of static observers, SO, whose spatial coordinates are fixed. They are characterized by velocity four-vector,

$$u\_{\rm SO} = \left. \frac{1}{\sqrt{f(r)}} \rho, 0, 0, \begin{pmatrix} \\ \\ \\ \end{pmatrix} \right| \tag{10}$$

#### 3.1. Travel time towards BH horizon

Let us consider the situation of observer A (Alice) whose frame of reference is in a radial free fall, l ¼ 0 towards the BH horizon (4). A's frame (or "spaceship") initially was at rest at a Mother Station, MS, located at r0. The coordinate time to cover the radial coordinate range ðr0;rÞ in this case is found from Eqs. (6)–(9)

$$t = -\int\_{r\_0}^{r} \frac{erdr}{\sqrt{\left(r - r\_g\right)\left[\eta\_g' - r(c^2 - 1)\right]}} \Bigg(\tag{11}$$

It diverges, t ! ∞ as A's spaceship approaches the horizon, r ! rɡ. The proper time, which is the time measured by Alice herself,

$$\tau = -\iint \left( \frac{e\sqrt{r}\theta r}{\sqrt{\left[\eta' - r(k^2 - 1)\right]}} \right) \lessapprox \tag{12}$$

turns out to be finite. This illustrates a manifestation of the most dramatic time delay: for distant observers (but actually for all observers exterior to the horizon), Alice's frame of reference would need an infinite time to reach the event horizon, while a finite time elapses for the co-moving observer, Alice herself. Another aspect of this outcome has already been mentioned. The speed, V, of the freely falling test particle as measured by a static observer, SO, follows from the expression (see also [5]),

$$
\mu\_{SO} \mathcal{U} = f \dot{t} \frac{1}{\sqrt{f(r)}} = \frac{1}{\sqrt{1 - V^2}} \tag{13}
$$

One finds then a general outcome: the speed of a test particle radially freely falling

$$V^2 = \frac{\mathfrak{c}^2 - f(r)}{\mathfrak{c}^2} \xrightarrow[f \to 0]{} 1\tag{14}$$

approaching the event horizon tends to the value of the speed of light in the vacuum. And this result is independent of the initial conditions. One may ask: how would that speed be changing inside the horizon? We discuss this question subsequently.

#### 3.2. Generalized Doppler shift: how to fix the instant of crossing of the Schwarzschild BH horizon

It is a well-known fact that due to the equivalence principle, an observer confined within a frame freely falling towards the horizon cannot identify the instant at which he/she crosses the horizon and if a black hole is large enough, such an observer would harmlessly cross the horizon without even noticing [6]. On the other hand, one can quite precisely determine that instant. How is this seeming contradiction possible?

Before resolving this, let us recollect a well-known result, that of the gravitational frequency shift. In order to do this, one considers radial signals of a fixed frequency, ω emitted at r0 (the location of the Mother Station) and recorded by a static observer at r > rɡ. The wave vector k of those radial light rays, k ¼ � kt ; ; kr ; 0; 0 � is (see Eqs. (6), (7)):

$$k^t = \frac{w}{f} \,' \tag{15}$$

$$k' = \pm \omega,\tag{16}$$

where � corresponds to out- and ingoing rays, respectively. If MS emits such a signal with frequency

$$
\omega\_{\rm MS}^{\epsilon} = \frac{\omega}{\sqrt{f(r\_0)}} \equiv \overline{\omega}\_{\prime} \tag{17}
$$

SO records it at r and measures its frequency as

$$
\omega\_{\rm SO} = \mu\_{\rm SO} k = \frac{\omega}{\sqrt{f(r)}} \tag{18}
$$

so

$$\frac{\omega\_{SO}^r}{\omega\_{MS}^t} = \sqrt{\frac{f(r\_0)}{f(r)}} \xrightarrow[f \to 0]{} \infty. \tag{19}$$

The frequency recorded by SO is indefinitely blueshifted: when r tends to rɡ, f ! 0.

When such radial signals are recorded by Alice, ωA ¼ uAk, at her instantaneous position at r, then she finds (see Eqs. (6)–(9) and (15, 16)

$$
\omega\_A^r = \frac{\frac{\omega}{\sqrt{f(r\_0)}}}{1+V} \tag{20}
$$

where V is the speed of her spaceship as measured by SO (placed at r) (see Eq. (14)).

Exchanging such signals, one can observe a (generalized) Doppler shift of the following form [7]:

$$\frac{\omega\_A^r}{\omega\_{MS}^\ell} = \frac{1}{1 + \sqrt{\frac{\epsilon^2 - f(r)}{\epsilon^2}}} = \frac{1}{1 + V} \tag{21}$$

and

$$\frac{\omega\_{\rm MS}^{\prime}}{\omega\_{A}^{\prime}} = 1 - V \tag{22}$$

The meaning of result (22) is as follows: signals coming from a frame infalling towards the black hole horizon are indefinitely redshifted (and ultimately disappear from the screens/ sensors)—such a journey seems to take infinitely long for external observers. This confirms our former conclusion. The result (21) on the other hand means that the Doppler shift of signals coming from MS allows Alice to identify the horizon quite precisely—the Doppler shift reaches a value of ½ on the horizon.

#### 3.3. Image collision or the 'touching ghosts' anomaly

With the speed of free fall tending to the speed of light in a vacuum, the generalized Doppler shift as characterized by Eqs. (21) and (22) and the dramatic form of the time delay in this case, this leads to yet another anomaly—image collision [8] or touching ghosts [9]. Signals emitted by Alice located within the infalling frame appear to get "frozen" in the proximity of the horizon (see, however, [10, 11]).

Let us consider another observer, B (Bob), whose spaceship also starts from MS, following Alice's spaceship. Alice and Bob exchange electromagnetic signals; how (when) does Bob perceive the instant of Alice's crossing of the horizon? The answer has been referred to as 'image collision' or 'touching ghosts' and it is as follows [8, 9]. Alice sends an encoded message: a signal that means 'I am crossing the horizon' (at the instant when her Doppler shift is half); Bob receives that message at the instant when he himself crosses the horizon. An interesting fact is that this effect, originally illustrated by means of Kruskal-Szekeres coordinates, may be interpreted in a general manner, without reference to any specific system of coordinates. Indeed, if Bob received such a message before crossing the horizon, that information would be transmitted to our part of the universe; this would contradict the fact that the horizon crossing can never be observed.

#### 3.4. Photon sphere

In the case of null geodesics in the equatorial plane, the wave vector components are as follows:

$$k^t = \frac{\omega}{f}, \; k^\rho = \frac{l}{r^2} \tag{23}$$

$$k'=\pm\sqrt{a^2-f\frac{l^2}{r^2}}\equiv\pm l\sqrt{\frac{1}{b^2}-V\_{\text{eff}}(r)}\tag{24}$$

where b is a so-called impact parameter. The function Veff r r ɡ 2 ð Þ¼ f is regarded as an 'effective r2 potential' for null geodesics (see Figure 1). The shape of null geodesics depends on the value of b. The deflection angle

$$\int d\varphi = \pm \int \frac{dr}{r^2 \sqrt{\frac{1}{b^2} - V\_{\text{eff}}(r)}}$$

Figure 1. Effective potential Veff r r ɡ 2 ð Þ ¼ f in the case of Schwarzschild spacetime (horizontal axis—r expressed in units, r2 rS � rɡ ¼ 2M).

� � is small for large values of b—light rays are only slightly deflected. It grows indefinitely as the impact parameter value tends to its critical value, b�1 ¼ Veff rphs cr . The impact parameter bcr corresponds to the so-called 'photon sphere' composed of circular trajectories, r ¼ rphs,

$$r\_{\text{phys}} = \frac{3}{2} r\_g \equiv 3M \tag{25}$$

which are (unstable) null geodesics:

$$\left(\left(\not{\theta},0,0,\,,k^{\rho}\right)\in\begin{array}{c}\frac{\omega}{f\left(\eta\_{\rm{obs}}\right)}\geqslant0,0,\frac{l}{r\_{\rm{ph}\_{\rm{0}}}^{2}}\right)\left(\begin{array}{c}\\ \end{array}\right)\tag{26}$$

#### 3.5. The shape of light cones

It should be noted that in approaching an event horizon, the shape of a light cone evolves in a characteristic manner. Indeed, observing radial in- and outgoing signals

$$ds^2 = fdt^2 - \frac{1}{f}dr^2 = 0\tag{27}$$

one finds,

$$\frac{dr}{dt} = \pm f \xrightarrow[r \to r\_g]{} 0\tag{28}$$

which may be illustrated as a sequence of vanishingly narrow cones.

#### 4. Interior of Schwarzschild BH

In order to describe the interior of the horizon of the Schwarzschild spacetime, one can follow an approach proposed by Doran et al. [12]. These authors showed that discussing the problem of an empty, but dynamically changing spacetime, one finds, using specific boundary conditions, the metric (4) of the interior of the Schwarzschild spacetime, that is,

$$ds^2 = \left( \left( -\frac{2M}{r} \right) d\left(t^2 - \frac{1}{\left(\frac{\mathcal{Y}}{r} - \frac{2M}{r}\right)} dr^2 - r^2 d\Omega^2 \right. \tag{29}$$

for r < rɡ ¼ 2M (see also [13]). This means that formally one can use Schwarzschild coordinates also for the interior of the horizon, but then one must remember about the exchange of the roles of the t and r coordinates. Inside the horizon, r plays the role of a temporal coordinate: it changes from rɡ to 0 and dr < 0; t plays the role of a spatial coordinate, changing between �∞ and þ∞ with dt taking both positive and negative values. The important consequence is a change of the symmetry of the system: instead of a static, spherically symmetric spacetime, one encounters a homogeneous, spherically symmetric and dynamically changing spacetime; energy is no longer conserved but (due to the homogeneity along the t-axis), appropriately, the t-momentum component is conserved.

Therefore, one can consider spacetime (29) as representing the interior of a Schwarzschild black hole. Accordingly, analogues of the phenomena described above outside the horizon will be analyzed.

First, one introduces a class of resting observers, RO, that is, those, whose spatial coordinates, t, θ,φ are fixed. Then, the velocity uRO four vector's only nonvanishing component is a temporal one,

$$
\mu\_{RO} = -\sqrt{-f}\,\mathrm{\partial\_r}.\tag{30}
$$

The class of infalling test particles located in Alice's frame of reference is described in the same way as given outside the horizon (Eqs. (6)–(9))—in this case, however, r < rɡ ¼ 2M, so f < 0. In this region, ingoing (�) and outgoing (+) null geodesics (that are planar) described as

$$
\int \frac{dt}{d\sigma} = \pm \omega \qquad \qquad r^2 \frac{d\phi}{d\sigma} = l \qquad \qquad \qquad \frac{dr}{d\sigma} = -\sqrt{\omega^2 - f\left(\frac{l^2}{r^2} + \eta\right)} \tag{31}
$$

differ from their counterparts outside the horizon by a small but important feature—the � sign is designated to a spatial coordinate, namely the r-coordinate outside the horizon and the t-coordinate inside the horizon. Having said this, one may now discuss specific effects (see [14, 15]).

#### 4.1. The speed of an infalling test particle

~ A test particle located in A's framework (Eqs. (6)–(9)), l ¼ 0, is freely falling FF. Then, a resting observer (30) measures its (squared) speed V2 as follows:

$$
\mu\_{\rm RO} \mathcal{U}\_{\rm FF} = -\frac{1}{f} \sqrt{-f} \sqrt{e^2 - f} = \frac{1}{1 - \hat{V}^2}.\tag{32}
$$

One finds then that (c.f. Eq. (14))

$$
\tilde{V}^2 = \frac{e^2}{e^2 - f}.\tag{33}
$$

This is, at first sight, a rather unexpected outcome: the speed is given by a formula inverse to the one obtained outside the horizon, Eq. (14). Another aspect of this result is revealed when one illustrates the speed outside and inside the horizon as measured by observers that are static, SO, and resting, RO (30), respectively (see Figure 2).

V2 V~ 2 particles in the Schwarzschild spacetime. The red curve corresponds to e ¼ 1, V2 ¼ r , the green curve to, e ¼ 0:5 and the Figure 2. Values of 'velocity' measured by SO (outside horizon) and by RO (inside horizon) of different test ~ rg blue one to e ¼ 0:2. The vertical line represents the horizon located at rg ¼ 2 (horizontal axis—r expressed in units M).

#### 4.2. The Doppler shift

Let us consider an analogy of the generalized Doppler effect inside the horizon.

#### 4.2.1. Frequency shift of signals coming from MS

#### <IP_ADDRESS>. Resting observers

One can start from an analogy of the gravitational frequency shift: a resting observer (30) records radially incoming signals coming from the Mother Station. Then, according to Eq. (18), the frequency shift is

$$\frac{\omega\_{RO}^r}{\omega\_{MS}^s} = \sqrt{\frac{f(r\_0)}{-f(r)}} \to \begin{cases} \Leftrightarrow & r \to r\_g \\ 0 & r \to 0 \end{cases} \tag{34}$$

One finds then that the gravitational frequency shift of the signals coming from MS and recorded by static, SO, and resting, RO, observers, outside and inside the horizon, respectively, as having a symmetric form with respect to the horizon itself (see Eqs. (19) and (34)).

#### <IP_ADDRESS>. Freely falling observers

The frequency shift of signals coming from MS and recorded by Alice, who is radially freely falling, is

$$\frac{\omega\_A^r}{\omega\_{\rm MS}^s} = \frac{1}{1 + \sqrt{\frac{\epsilon^2 - f}{\epsilon^2}}} \to \begin{cases} \frac{1}{2} & r \to r\_g \\ 0 & r \to 0 \end{cases} . \tag{35}$$

Expression (35) is the same as its counterpart outside the horizon (21): it turns out that the frequency shift is a continuous and decreasing function from 1 to 0 during the trip through the horizon; as emphasized earlier, with the factor 1 marking the horizon (see Figure 3) 2

ωr Figure 3. Monotonic and continuous change of the frequency ratio A (redshift—Vertical axis) outside and inside the ωs MS horizon (horizontal axis—r expressed in units M, rɡ ¼ 2).

#### 4.2.2. Frequency shift of signals inside the horizon of BH

One can consider the exchange of signals by observers at rest inside the horizon. One can distinguish two types of signals: going along the direction of homogeneity, that is, the t-axis, and signals propagating perpendicularly to this axis.

#### <IP_ADDRESS>. Signals propagating along the t-axis

The frequency shift of signals exchanged by two observers at rest at t1 and t2 depends on the emission instant, r1 (recording instant r2 is fixed by the distant t1, t2):

$$\frac{\omega^r(t\_2)}{\omega^e(t\_1)} = \frac{\sqrt{-f(r\_1)}}{\sqrt{-f(r\_2)}} \xrightarrow[r\_2 \to 0]{} 0. \tag{36}$$

One finds that in this case, the frequency redshift tends to zero at the ultimate singularity (see Figure 4).

#### <IP_ADDRESS>. Signals propagating perpendicularly to the t-axis

The wave vector of signals propagating perpendicularly to the t-axis has two non-vanishing components k ¼ kr ∂r þ k φ∂φ (because of the planar character of the trajectory, one can choose π θ ¼ 2, i.e. the equatorial plane). Then, the frequency shift, for two static observers placed within this plane perpendicular to the t-axis, is given by

$$\frac{\omega^r(\varphi\_2)}{\omega^{r\text{\textquotedblleft}}(\varphi\_1)} = \frac{r\_1}{r\_2} \xrightarrow[r\_2 \to 0]{} \infty. \tag{37}$$

One finds an indefinite blueshift at the ultimate singularity (see Figure 5).

Figure 4. Frequency redshift (vertical axis) for the signal propagating along homogeneity direction between instants, rB ¼ 0:9rɡ ¼ 1:8 and rA ¼ 0:2rɡ ¼ 0:4 as a function of r (horizontal axis—In M units, rɡ ¼ 2).

Figure 5. Frequency blueshift (vertical axis) for the signal propagating perpendicularly to the homogeneity direction, between instants, rB ¼ 0:9rɡ ¼ 1:8 and rA ¼ 0:3rɡ ¼ 0:6 as a function of r (horizontal axis—In units M, rɡ ¼ 2).

#### 4.3. Photon sphere analogue

Null geodesics propagating perpendicularly to the t-axis resemble trajectories belonging to the photon sphere. Indeed, they are determined by the wave vector,

Black but Not Dark 55 http://dx.doi.org/10.5772/intechopen.77963

$$(0, k', 0, , k^{\wp}) \equiv \quad 0, -\sqrt{\frac{-1}{f}} \frac{l^2}{r^2}, 0, \oint\_{\Gamma} \left( \right.\tag{38}$$

having only one spatial component, the angular component, kφ corresponding to a circular-like motion. There is one significant feature distinguishing these null geodesics from the circular trajectories of radius r ¼ rphs outside the horizon: they circulate over a sphere of an everdecreasing radius. One can see from the null condition:

$$-\frac{1}{f}dr^2 - r^2d\phi^2 = 0\tag{39}$$

that the rate of change of the radius of such a sphere is proportional to r, which is a temporallike (decreasing) coordinate.  Outside the horizon, a light ray belonging to the photon sphere can (in principle, as it is a circular trajectory of unstable equilibrium) unwind infinitely many times. One can ask then: inside a black hole, how many times can a light ray orbit along a photon sphere analogue before reaching the ultimate singularity?

The answer to this question is quite unexpected: it is exactly a single half rotation.

Indeed, by using Eq. (39), one obtains

$$
\Delta\varphi = \int \frac{dr}{r\sqrt{\frac{-f}{r}}} \overline{\int}\_0^{2M} \frac{dr}{\sqrt{r(2M-r)}} \overline{\int}^{\pi} \pi \tag{40}
$$

This means that the angle traversed by a light ray is equal in this case to π. A general property is that the deflection angle for a light ray within the BH horizon cannot exceed π.

### 5. The horizon of a Schwarzschild BH

Among various interesting properties of the Schwarzschild BHs horizon, there are at least two that are relevant to our discussion.

The first relates to the speed of an object crossing the horizon. As described earlier, the value of the speed of Alice's spaceship tends to the value of the speed of light c as it approaches the horizon. Does that speed take the value c on the horizon? There are no observers residing on the horizon, but other observers, crossing the horizon, would in principle be able to perform such a measurement. Performing this kind of thought experiment, one obtains the following: the speed of Alice's spaceship crossing the BH horizon is less than the speed of light. The value of that speed depends on the initial conditions.

The second is linked to any outgoing light ray trapped at the horizon. It may be a signal emitted by Alice at the instant she was crossing the horizon with the encoded message: 'I am crossing the horizon now'. If it was a signal of some specific frequency, what would be its frequency as recorded by Bob, when he crossed the horizon? It turns out that such a signal 'ages': it is redshifted and the value of the redshift becomes greater as the original distance between Alice and Bob increases [15].


**Author details**

Vu T. Tan\* and La The Vinh School of Chemical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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