The Black Earth

For an insight into the curvature of space due to the gravitational field, as it was predicted by Albert Einstein in his theory of General Relativity, it is suitable and impressive to see the effects of curved space act on well known objects. In comparision with the usual views then one may get an intuitive feeling of what `curvature of space' really means.

Planet Earth is quite a well known object to all earthlings, and therefore this simulation series will demonstrate the appearance of Earth, if it became a black hole from one day to another.

As an starting point one needs a `virtual laboratory' with planet Earth inside. Even better, the lab shall contain the Moon also, such that the scenery isn't too trivial at all. In nature the moon resides at a distance of about 60 times the radius of Earth, but in the virtual lab the experimentator may `play the role of God' and set the Moon to a distance of a mere 1.5 times the radius of Earth. This distance is called the Photon Orbit. At the Photon Orbit light, i.e. photons, may rotate around the Black Hole on an perfect circle. The visual effects resulting out of this behaviour will be shown in the following images and descriptions. To improve realism of the situation the `lab desktop' is preferably to be eliminated from the scene. In reality the system of Earth and Moon is embedded within the star-filled celestial background, and accordingly in this next simulation image the Moon is shining above the nocturnal central parts of Africa in front of the starry sky. Its data were provided by the SAO catalogue of the 258.000 brightest stars (compiled by the Smithsonian Astrophysical Observatory). Data from the Earth's surface (day and night part) came from Living Earth ®, the Moon's surface has been constructed using data from the US space probe Clementine. The scene could be rendered even more realistic if the earth's atmosphere is taken into account also (which is enlarged here by a factor of 15 as compared to its true extension). Please note the appearance of the atmospheric reddening at the poles near the terminator, demonstrating the aurora as seen from space. Unfortunately the addition of atmospherical effects increases computation time by a factor of 60. Therefore the succeeding simulations have been made `in vacui'. Now, what happens, if Earth had such an mass, that it would enclose 90% of its Schwarzschild radius? This percentage is near stability limit of a static star; the exact value is 9/8 of the Schwarzschild radius, which corresponds to a fill rate of 88%. If any object is compactified within a volume smaller than this limit, it will inevitably collapse even further due to its own gravitational field, finally ending as a Black Hole. The gravitational field of such an compact object is of such an intensity that light rays are `bended downwards' towards the mass. A light ray, which in flat space would pass straight ahead over the north pole, would be attracted by the gravitation and `fall down' to the the north pole. Therefore the north pole - and even the regions beyond - become visible for an observer residing in the equator plane. The same happens to the south pole, which becomes visible also, contained within the same view of the north pole. Similar conditions are valid at the equatorial plane: The light is bent around the Earth and therefore the full day side can be seen at once together with the full night side; moreover beyond the night side the day side becomes visible again, since the light has made one `turn around the Earth', and similarly somewhat right of the day side the night side is visible for the second time. Effectively one is able to watch the complete surface of a Black Hole at once (and even more than once). But to be correct it has to be stated that the simulation shown here is wrong in some aspect. If it is possible to see the entire surface from any location, then it is also possible to see the entire celestial sphere from any location on the surface. Therefore the sun were visible from any surface location, because the light coming from the sun would also be attracted to parts of the Earth's surface which are unreached by the sunlight in flat space, i.e. the former nocturnal hemisphere. The distinction between night and day becomes meaningless under such conditions, the sun is shinig everywhere at the same brightness. This influence of the space curvature to the infalling light was not taken into account for the simulations presented here, on the one hand, because this were quite complicated to be done computationally, on the other hand, because this method of - `false' - visualization enables a more intuitive and impressive view of the difference between front side and back side. Now what would an observer see, who is residing within the curved space, not just looking at it from outside at save distances? This image shows the view to the moon, seen from a radial distance from Earth of 1.1 radii, i.e. .1 Earth radii above the surface (ca. 600km). The strange look is obvious: the Moon seems to be stretched to become an elliptical shape, the starry sky also appears to be somewhat `crunched' near the horizon - as stated earlier, one is able to see the complete sky from any point on the surface, because the star light is spiralling inwards from any direction. Moreover the entire surface is visible from any point within the surface itself, since light rays, which can't escape from the surface due to their zenith angles beeing too large, will be attracted back to the Earth's surface. For that reason the sunward side (`day side') appears behind the night side near the horizon, somewhat below the Moon. But in a virtual lab there is no reason counting against the idea to have the mass of Earth beeing increased up to the Schwarzschild limit. A Schwarzschild radius of about 6000km, as it is the Earth's radius, corresponds to about 2500 masses of the sun (which by itself has an Schwarzschild radius of about. 2.5km), which had to be compressed to fit within the volume of the Earth. Such an object isn't physically stable any more, nethertheless it can be simulated in spite of that. The appearance compared with the former simulation isn't very distinctive, the bending tr the horizon becomes somewhat stronger, also becomes the distorsion of the moon and the starry sky. However one should note that this is one of the last seconds in the life of an observer falling into a Black Hole. From this location he might still escape with some powerful spacecraft (Warp drive should do it), but a couple of steps towards the center, and any chance to escape gets lost forever. There is no way to leave the region within the event horizon (within the Schwarzschild radius). The `last glimpse' to the Moon could look like this image: The falling astronaut (or kosmonaut) has approached the event horizon as near as mere 10km. Similarly some mountaineer on a somewhat heightened Mt. Everest within the central Sahara could see the same (if one neglects the influence of special relativity effects due to different motion of these two observers, i.e. the astronaut should not be in free fall, but somehow rest at a constant radial distance). But now let's go back somewhat outwards, up to the more interesting regions around the Photon-Orbit, which was mentioned already. At first impression this position doesn't appear very exciting. Neither the view towards the Moon, nor the view in the opposite direction seem to be very particuliar. Again, the stars become more and more distorted near the horizon, and, as before, the day side is visible beyond the night side and vice versa, which is a hint to the strong curvature of space at this place, but it doesn't look too interesting at all. Could this really be the region of rotating photons? But there is a solution to stop disappointment because of this `boring' appearance: Let's play God again and set an artificial torus around the Earth, just at the radius of the Photon Orbit. One may consider this torus as some kind of `hyperfinanced space station' (it's a simulation!). The Moon, residing at the Photon Orbit also, becomes therefore be `pierced' by this torus. Before `switching on' gravitation, it's better to go to some greater `safety' distance, say, seven times the radius of Earth. But now: `Click' - gravitation switched on. Light falls down on large spiral-like paths, is inevitably beeing attracted by the Earth's mass and the whole planet appears to be of an size which even extends the current field of view. Space curvature changes distances, and so it does change the distance to the planet's surface. However a spaceship orbiting the planet at the same speed and at the same distance as in the previous image (in flat space) would need exactly the same time to do so - the circumference at a certain radial distance doesn't change at equal Schwarzschild radial distance. It is just this property of the Schwarzschild radial distance which defines it. Nethertheless the true (metric, measurable) distance to the planet's surface is changed by the space curvature. To have a better overview it is required to use another camera lens, e.g. one with an horizontal view angle of 120° instead of 50° as before. So the full distorted surface and the near space arround fits into the view field. Again, the entire surface of the planet is visible. But also the torus, which is wound around the Earth at the equatorial plane, is visible at each point of its course. The torus seems to be laid around the north pole. This is easy to understand, since the observer is residing somewhat above the equatorial plane and light rays originating from the torus are bent downwards towards the north pole, finally entering the observer's eye.

But now let's come closer to the scenery. Our spaceship shall reside in a distance of two times the Earth's radius, somewhat above the equatorial plane. The view to the torus seems to be quite common in the near area, but instead of disappearing behind the horizon, as one would expect it when looking at our real Earth and an imaginary torus around it, the torus is suddenly going upwards, running around the north pole and coming back from behind. Unfortunately the view field is too small to cover this full behaviour and to show the north pole with its surrounding torus in one image, but with some experience from the previous images this behaviour is clear. As it was mentioned already: The entire surface of the planet can be seen from any position, as it were a flat disk. Hmm. Does this mean that people from the dark Middle Ages were right in their view of the world?
Ok, if the space ship is hoovering somewhat above the equatorial plane, the torus appears to be bent northwards, and this would happen analogeously when the space ship is positioned below the equatorial plane. Then the question arises: What happens, if the spaceship is placed exactly at the equatorial plane? Then - because of the symmetry of the situation - the torus can't go either upwards nor downwards...? But effectively a torus isn't an infinitely small line, some parts are above the equatorial plane, other parts are below. Therefore the torus seems to be `split', it seems to be stretched along the whole horizon and to fill the entire horizon. Since the torus by itself is an object of relatively small size as compared to the horizon, it becomes very distorted. This effect visualizes one more property of the Photon Orbit: Circular orbits along the Photon Orbit aren't stable, but an instable path - one small perturbation within the exact circular orbit enforces the light to either spiral inwards into the Black Hole or to escape to infinity.
But now the space ship shall take a stop exactly at the Photon Orbit, somewhat above the torus, such we have a tangential view along the torus. And, as it is to be expected: The torus, which truly circulates the Earth, appears to be a straight line, since the curved path of light rays is parallel to the curved torus everwhere arround the Earth! Of course at some distance the torus seems to be bent northwards, as it is known already from the former simulations. The straightened torus can't disappear behind a horizon formed by Earth's surface, as it would be the case in flat space, since there is no more such kind of horizon when the entire surface is visible. On the other hand the torus also can't get lost in infinity, as it were expectable with an infinite bar in flat space, because the torus is a finite object of finite length.
Next flight stop is a position just within the Photon Orbit, at an radial distance of 1.4 times the Earth's radius (which is identical to the Schwarzschild radius). The torus still looks like a straight object, it even seems to be curved somewhat outwards. Near the Earth's horizon, beyond the dark areas of the night side, there appears a bright region, which is a hint, that the day side becomes visible here after one `turn around' of the light - some hint that we are residing in some part of even greater space curvature.
Now, at the last stop, a view from a position just 1% outside of the Schwarzschild radius is shown. Both the Earth's surface and the torus appear to be curved outwards, just as they would `collapse' soon. We are near the region `of no return' (does this imply that the C/C++ programming language can't exist there...?).
But now let's jump back to the Photon Orbit. If light really can run there on circular orbits, one should be able to see his own back, right? There are some - minor(?) - problems with rendering the observer and his back within computer generated simulations, and therefore it is easier to use the Moon as `observers back'. The Moon is still within the scenery and its position at the Photon Orbit shouldn't be without reasons. But for now we concentrated on the torus and the Moon wasn't visible in the tangential views, since it resides just behind the observer. And indeed, if we eliminate the torus, which hides the direct view to the Moon around the Earth, then far away there appears a dim grey stripe along the horizon, which effectively is the extremely stretched back side of the Moon.
This isn't very impressive. The Moon's structure doesn't contain enough detail of high contrast such that anything could be recognized in the photon-orbit distorted appearance. There is no definite distinction to some bright area at the Earth's surface itself. Once again, this requires the experimentator to play the role of God and insert a light source like a small shining reddish sphere into the scenery. If positioned onto the backside of the Moon, it becomes a somewhat more complex structure. In the flat space, this scenery looks like this image.
The same situation as before, photosimulated in flat space, but using an horizontal view angle of 10°.
Now the Moon and its small lamp is positioned just behind the observer and gravity is switched on again. Using an horizontal view angle of a mere 1°, which is 1/10th of the previous view, one may again see the long gray stripe near the horizont, as before, but now the red lamp becomes visible also, the latter beeing stretched to be a long stripe also.
One more zoom, using an horizontal view angle of 0.1°. The distorsion due to the light `turn around' along the Photon Orbit is too big to leave any details, but there are no doubts left that it is the the small red light source at the back side of the Moon, which is visible here `around the Earth'. So this is how an observer residing at the Photon Orbit would see his back side (please compare this view to the analogeous 10° view in flat space above).
Well, at last: if indeed light is able to rotate along the Photon Orbit, then it should be possible to see the Earth's surface arbitrary times, because light may take a turn around the Earth more than once until it comes down touching the surface. Unfortunately the Moon itself hides the view path along the Photon Orbit in the previous images. Therefore we should set the Moon onto a somewhat more distant orbit and - voila! - beyond the night side, beeing itself behind the day side, there appears the day side again. By estimation it is Indonesia, which is hereby seen the second time, possible because of a complete Earth turn around of the light. The first time Indonesia is visible just below the observer, as it is shown in the previous images (like the tangential view along the Photon Orbit). This first image is located at nearly the position were it is to be expected in the flat space scenario. However, the view field in this image is only a very small section of 1° along the horizon. If one would select even smaller view fields, one could see even more repetitions of the same geographical areas, each repetition stretched more than the previous one by large amounts, and each one beeing lots smaller than the preceeding appearances. As mentioned earlier, the Photon Orbit is an instable circular orbit.

The End

* The Day when Earth became a Black Hole: In some manner, this day was July 20th 1997, since this was the day when most of these simulations have been created.