Written in Dec'17, and translated from [[Refleksjoner om relativitet i og rundt sterke gravitasjonsfelt|the original Norwegian]] Jan'24. ## Reflections on Relativity[^1] In and Around Strong Gravitational Fields ### Abstract We[^2] discuss the perspectives of observers in different inertial frames in and around strong gravitational fields. As an example, we investigate in detail the situation when one of two observers are falling into a black hole. ### 1 – Einstein's General Theory of Relativity In 1905 the situation was as follows: Newton's law of universal gravitation — one of the most successful descriptions of nature of all time, a simple equation, $\qquad\qquad\qquad\qquad\qquad\qquad\qquad F = G \frac{m_1 m_2}{r^2}\, , \qquad\qquad\qquad\qquad\qquad\quad (1)$ able to describe with great precision everything from the solar system, to the relationship between the moon and the tides (until that point an observed but unexplained correspondence), to *why things fall to the ground* — was incompatible with Einstein's newly published theory of relativity. The problem was that Einstein's theory of relativity did not allow for two objects to affect each other instantaneously: the time from action to reaction had to[^3] be greater or equal the distance between the objects divided by the speed of light[^4]. In Newton's theory, changes to the gravitational field had an immediate effect on objects within. It took him ten years to discover, but the solution to the problem, published by Einstein himself in 1915, would turn out to be both elegant and deep. Though it quickly made a name for itself as both unintuitive and mathematically challenging[^5], the central equation[^6] in *the general theory of relativity* is easy enough to write down: $\qquad\qquad\qquad\qquad\qquad\qquad R^{\mu\nu} - \frac{1}{2}Rg^{\mu\nu} = \frac{8\pi G}{c^4}T^{\mu\nu}\, . \qquad\qquad\qquad\qquad (2)$ The essence of the equation, and the essence of general relativity, lies in a re-interpretation of time and space: they were no longer constant, unchanging, eternal, like they had been treated since the time of philosophers[^7], but were now *bendable, stretchable*, constantly evolving, like a stretched-out sheet covered in marbles; like a lake in a gentle breeze. Many found this hard to accept, so the theory was (of course) met with enormous skepticism by the establishment[^8] — ironically first and foremost by philosophers[^10], who thought themselves qualified to comment on the matter. So it took ten years to discover the correct equation (2). If you were to ask Einstein in 1915, he would have answered that finding a non-trivial solution[^11] to the equation might well take another ten years. It took a month. ### 2 – Schwarzschild's Solution Karl Schwarzschild's[^12] story is a tragedy worthy of the greeks: an unknown genius that over the course of only a few weeks — while serving on the Russian front during World War 1 — solves what could easily have become the great mathematical puzzle of the decade; the discoverer of an equation with greater and stranger consequence than anyone at the time realized — than anyone *could* have realized; a physicist on the brink of international renown, central in the discovery of the greatest scientific revolution since Isaac Newton. In a letter dated December 22 1915, he presents his solution to Einstein: $\qquad\qquad\quad ds^2 = -(1 - \frac{r_s}{r})c^2 dt^2 + \frac{1}{1 - \frac{r_s}{r}} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)\, . \qquad (3)$ Einstein: «I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy[^13] with a few words of explanation.» Five months later he was dead, taken by a rare autoimmune skin disease that he had brought back from the front. He was 42 years old.[^14] The equation describes how the gravitational field of a planet or star or any static object at all, affects another object, for example a person[^15], as observed by *another* person *from afar*. She is falling; he sees her falling[^16] ... and her fall is described by (3). Look at the equation one more time. In primary school we learned that you can't devide by zero, but in a scenario where the center of the gravitating object, $r$, is equal to the so-called *Schwarzschild radius*, $r_s$, this is exactly what we do: we divide by zero. Try to see it for yourself. It happens in the second term to the right of the equality sign. In technical terms, this is called a *singularity*. The formula no longer makes physical sense in such a scenario. In Einstein and Eddington and Silberstein and Schwarzschild's time, it was simply assumed that such a scenario could never manifest. This is because in most cases, the Schwarzschild radius sits *inside* the gravitating object, and there the equation is inapplicable anyway. There, one must use different solutions to (2)[^17], and these solutions contain no singularities. For the Schwarzschild radius to sit *outside* the object, the object would have to have an *enormous* density[^18], and it was simply assumed that such huge densities were physically impossible. Today we know better. Today we know that not only do such objects *exist*, but that they are *numerous*. Today we know that such objects are central to the development of our universe, from The Big Bang until today. Today we know that almost all galaxies, including our own, orbit around such an object. Today we know such objects as *black holes*. ### 3 – I Orbit a Black Hole, and I see her falling. Nothing, not even light, can ever escape a black hole. When something has passed within the Schwarzschild radius, the so-called *event horizon*, there is no way back. There is another singularity in Schwarzschild's equation; another case where we find ourselves dividing by zero: when $r = 0$. That is, at the black hole's centre. There sits an infinitely tiny object, a point of infinite density, and possessing *the entire mass of the black hole*.[^19] Everything that has ever fallen into the black hole has ended up here, and for every sun it swallows, every planet it devours, every object, big or small[^20], it gets heavier — but it remains a point. There is no avoiding it, and there is no way back. I see her falling. She is falling, down towards the event horizon, down towards the Schwarzschild radius, closer and closer and faster and faster. I see her looking up at me, but in her eyes I see nothing, just emptiness. For a brief moment I think that she *wants* this, that she is welcoming it, the event horizon, the Schwarzschild radius, the singularity, the infinitely tiny point below that weighs more than one hundred sextillion times as much as her; for a moment it looks like she's *longing* to become a part of it, compressed to a point smaller than any of the atoms that make up her body; but no, nobody wants a destiny like that.[^21] Then I see her ... slow down. At first it makes no sense: gravity *accelerates* objects. Always. And there is no resistance out here, nothing that can slow her down. Why is she slowing down? But then I remember: the singularity. At the event horizon, we are dividing by zero. And you can't divide by zero. The event horizon can not be reached, and it can not be crossed, because then we would be dividing by zero and we can't divide by zero. So she slows down, slower and slower, eyes turned towards me, empty, unmoving, until it looks like she is completely still. Her colours start to shift towards red. Gravity is strong down there, and her light is struggling to reach me, and at that point it shifts towards red. Soon she will be invisible, light stretched deeper into the inrared, until only radiowaves remain ... but she will still *be* there, frozen on the surface, at the *event horizon*. It can not be reached, and it can not be crossed. I still see her. Maybe it is not too late. Maybe, if I am fast enough, maybe I can reach her. If I turn around, jump after her, reach out my arm, stretch out my fingers, maybe I can reach her. Maybe it is not too late. ### 4 – I am Falling Into a Black Hole, and I see him reach out for me. I think I must be falling, faster and faster, because he is getting smaller and smaller, but it doesn't *feel* like I am falling. It looks like *he's* the one falling, as if everything *but* me is falling ... but I know better. I am falling into a black hole, and I am falling faster and faster. I look ahead. The stars are moving in unnatural patterns, like fireflies orbiting a well, but I know better. Light is affected by strong gravitational fields, gets bent around the black hole, and the light that reaches me will be more strongly affected the deeper I fall. A distorted view of reality ... but it is not *real*. The well is getting larger. The darkness is strangely beautiful. Never has anyone seen a deeper black. It is stretching around me now, embracing me, as if it has been waiting for me, ever patient, like a mother embracing her child. Ashes to ashes. Singurality to singularity. The stars are gone. I trace the edge of the darkness with my eyes as it closes in around me, watch it close off the rest of the universe, until all that remains is he. He is distant now, like a figurine framed by the entire universe, a shiny frame on a black wall. It looks like he is reaching for me. As if he is about to jump in after me — for a brief moment I think he is about to jump in after me — but I know better, *he* knows better, he knows it's too late. He would never sacrifice himself for something he knew was hopeless.[^22] He disappears, and darkness embraces me. All is blackness. No light can reach me anymore. I am falling; I *know* I am falling, in towards the singurality, in towards infinite density, in towards the tiny terminal station at the end of the line ... but it doesn't feel that way. It feels like I am being perfectly still, like I am only hovering in infinite darkness. Is this the inside? Am I inside the black hole? I assume so. And there is no way out. There is no way out. ### 5 – Time and Simultaneity and a Theory of Relativity I am falling, down, down towards here. Loosened the strap and stepped off the craft. Carefully navigated myself by exerting hydraulic fluids, tiny pressure engines fastened to the space suit. I still see her; she looks red and unnatural, and soon she will be invisible, but she will still *be* there, frozen on the event horizon. ... Right? I am falling down towards her, but now that I am falling I could swear that she is no longer standing still; no longer frozen in time. What is happening? What happened with the singularity? The event horizon that could never be crossed, at least not from my persp— I changed my perspective. How could I be this dumb? I changed my perspective when I exited stable orbit. Did I really think that she would still be there, frozen in time at the event horizon, and that it would be for me only to head on down and pick her up? Now I am free falling too, deeper and deeper into the gravitational well, just like her, and now for me, just like for her, spacetime is described by a different equation altogether: $\qquad\qquad\qquad\quad ds^2 = -c^2 dt^2 + dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)\, . \qquad\qquad\quad (4)$ The equation for flat spacetime. Good old special theory of relativity from 1905, he discovered it when he was 26, I am 26; no singularities in this equation, nowhere to divide by zero. How could I be this dumb. She is falling faster and faster, into a black abyss. It's been a long time since she fell[^23], and I only stood there and let it happen, too paralysed to do anything. But it will only have been a couple of seconds for her. Time is relative. Simultaneity, too. Conditional validity; relationarity. She is still alive. She is still alive — and she is disappearing. Her light can't reach me anymore, can no longer resist the gravitational pull, and she is disappearing. It is too late. Nothing, not even light, not even an atom, not a single photon, can ever escape a black hole. When something has passed the event horizon, it is too late. It is too late for her ... but not for me. I can still turn around (I think) — how much hydraulic fluid is left in the spacesuit? Enough (I think; I hope). I can still turn around ... without her. Without Victoria. I make up my mind. Pull the triggers. The engines activate — I feel their vibration through the suit, hear their hum through my body, and I feel myself accelerating. It feels like gravity. I smile at the irony. Close my eyes and imagine myself lying on the ground, on a grassy field, sun above me and birdsong around me and a gentle breeze in my hair. ### 6 – There is No Way Out. She is lying in bed. Awake. Eyes open, fixed on the ceiling. She doesn't react when I say her name. It's been like this for a week now. I know she can hear me, and she eats the food I make her, and she drinks the water I bring her, but she doesn't react when I speak to her. Five days since she last looked me in the eyes; seven days since I last heard her voice; ten days since she last kissed me. Times like these come and go and come back again — but this time seems different. It is like she is frozen in time. Like she is orbiting a black hole. And I hover without, watching it happen, watching her freeze in time at the event horizon, watching her colours shift until I barely recognize her, paralysed, unable to do anything. Until I did something. Such darkness is not unfamiliar. I have orbited black holes before. I know what it's like to go dark for days at a time, only to resurface and wonder where all the time went ... but as we have learned: time stops in and around strong gravitational fields. I laid down next to her. Didn't say anything, didn't bring food or water and didn't try to get her to eat more, she must not forget to eat, she is looking so skinny; didn't try to reach her, didn't say her name, didn't say hey how are you doing. I laid down next to her, on my back, like her, eyes fixed on the ceiling, like hers. Then came the tears. I didn't realize it was going to happen, but I should probably have known. It started out a swelling in the throat, a painful tumour that refused swallowing. Then it grew larger, and stronger, and the effort of holding it back made my eyes well up; the tumour was suffocating me, I couldn't breathe, it kept growing, I couldn't hold it back anymore. I gasped for air. Held my breath. Tried to swallow the tumour, this big swollen apple that somehow had sprouted in my throat. Failed. Gasped again. And again. Soon enough, tears were flowing. I cried, as I haven't cried in years, and it felt ... good. Right. Deserved. It was then that she turned towards me. Looked me in the eyes, for the first time in five days, her own eyes welled up with tears. She pierced deep into my eyes, deep into my soul, and for the first time in ten days, I saw *her*. Victoria. She is still here. She is still alive. She may have crossed the event horizon, we may both have crossed the event horizon, and there may be no way out, but we are still here. Together. Alive. She put her arm over my chest; laid her head on my shoulder; closed her eyes; cried. We cried. We held each other, and we cried. #### 7 – The Way Out A light. Far away in the distance, deep in the blackest darkness. It is unfocused, wavy, as if I am looking through water; it must be gravity affecting the light. It is strong now, the gravity — strong enough that I can feel it pulling on my body, as if someone is gripping my ankles and pulling me down. Faint. My heart is struggling to resist the forces; my blood is being pulled downwards too, and blood pressure in my head is dropping. I feel faint. I don't know how much longer I can hold onto consciousness. But I see a light. Far away in the distance, deep in the blackest darkness. Could it be...? The light is approaching, and I see him, it *is* him. His face, illuminated by his helmet, is swimming before my eyes — I no longer know if it is the failing blood pressure or the play of gravity. I don't care. He navigates towards me, turns off his engines, falls, we are falling, side by side. His hand is reached out. I try to lift mine, but it is hard. I can feel consciousness failing. I try again. Muscles feeling foreign, as if it someone else's hand that I am lifting, but I do it, I see it move. Up. Out. It reaches his. Grips. He pulls me towards him. Holds me. I can see him clearer now. He is crying. Smiling. «It is going to be okay,» he says. His voice is distorted over the radio, his words hard to pick up. I don't believe him. ... But it is good to hear his voice. It is so good to hear his voice. It is so *infinitely* good to hear his voice. I can feel tears welling up. Won't bother holding them back. Maybe it is time. Maybe it is right. «It is going to be okay,» he says again. «I have found a way out.» I open my eyes. Look at him questioningly, as questioningly as I can manage with swimming eyes. «I have read about a theory ... some people believe that in the centre of some black holes, there is no singularity, no infinite density, no infinitely small point.» He sounds out of breath, as if every word is a struggle. «Some people believe that in the centre of some black holes, there is ... a kind of ... *portal*. That they're just portals, to a different place ... maybe even a different time. Black hole on one end ... *white* hole on the other.» «How,» I start. It is difficult to push words out of my lungs, as if the air itself is heavier now. My voice sounds strange, foreign, someone else's. I don't recognize it. «How do you know ... that *this* one ... is a portal?» «I don't,» he says, his voice barely audible now. He smiles again. «But I choose to believe it.» «It is going to be okay,» he says, and he holds me. He cries. We hold each other, we fall, and we cry. «It is going to be okay. Just wait and see.» [^1]: Freely translated from the Norwegian dictionary: relativity (noun): being relative; conditional validity; relationarity. [^2]: I. It's only me here. [^3]: (has to) [^4]: $c = 299$,$792$,$458$ m/s [^5]: See *e.g.* this entertaining exchange that is said to have taken place between one Ludwik Silberstein and Sir Arthur Eddington, the latter one of the most important figures in the history of relativity, at a lecture a few years after the publication of the theory: Silberstein: «Professor Eddington, surely you must be one of only three people in the world that truly understands relativity!» Eddington: (hesitates). Silberstein: «No need to be shy, Eddington.» Eddington: «Oh, no! I was wondering who the third one might be!» [^6]: Or rather, *equations* — $μ$ and $ν$ are in the following equation four-dimensional vector indices, which means that we really have not just *one* equation, but *sixteen*, all dependent on each other. [^7]: (the *Greek* philosophers, that is) [^8]: It is worth bearing in mind that Einstein was only 36 years old in 1915, though already a highly regarded physicist — his big breakthrough had come with the publication of the *special* theory of relativity, ten years earlier. He was 26 at that time.[^9] [^9]: (I am 26.) [^10]: (not necessarily Greek) [^11]: where the *trivial* solution is the case with zero gravity. The theory then reduces to the good old special theory of relativity from 1905. [^12]: Black *shield*, not black child. [^13]: *Königlich-Preußische Akademie der Wissenschaften* — The Prussian Academy of Science. [^14]: I wonder whether Schwarzschild could have beeen the third person Silberstein had in mind. Eddington would of course be well acquinted with both Schwarzschild's work and his untimely death. [^15]: (for example Victoria) [^16]: (I see her falling) [^17]: (of which the first and simplest was *also* discovered by Schwarzschild in 1915) [^18]: In case you're wondering, the Schwarzschild radius is give by $r_s = \frac{2MG}{c^2}$, where $M$ is the mass of the object, $G$ is Newton's gravitational constant, and $c$ is the speed of light. Given that $G$ and $c$ are constants, this means that the Schwarzschild radius is given by the mass of the object alone. The Schwarzschild radius of Earth is nine millimeters. [^19]: This mass is rarely trivial: the black hole at the centre of our own Milky Way is more than four million times heavier than the sun, which is more than three hundred thousand times heavier than the earth, which is more than sixty sextillion times heavier than me; more than one hundred sextillion times heavier than *her*. [^20]: or 173 cm tall and weighing 60 kg [^21]: Do they? [^22]: Would he? [^23]: (ten days)