maybe an inexact quote from a marilyn manson song will clear things up...
the concept of god is one thing. i can entertain that to some degree.
the concept of god as in christianity or other self centered faiths is quite another kettle of fish.
Show off your scientific background here 2 : The return
- Thread starter Frostblood
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maybe an inexact quote from a marilyn manson song will clear things up...
the concept of god is one thing. i can entertain that to some degree.
the concept of god as in christianity or other self centered faiths is quite another kettle of fish.
actually, im more or less familiar with this theory.
and just cos i beleave it to be rather circular in its logic, (and while i do enjoy religious debating) that doesnt mean i dont understand and apreciate your theory either.
but just because you measure time with light doesnt mean that it is dependant. wasnt it you who answered an earlier question of mine about why light was so important? its just cos its about as fas as you can go isnt it? well, in that case why does there need to be light for there to be time?
the big bang created matter yes, and maybe even light too, but as far as i understand, isnt the prevalent theory that it was created from a vast compression of energy? not disimilar from the whole black whole thing? only when it became compressed enough it forged into matter. in a great explosion and so on.
where was i? oh yes, light and time and before the big bang. nothing existed before the bag then? except for the energy that caused the big bang? well, just because you cant measure something doesnt mean it wasnt there. surely there was something before the big bang, otherwise there simply couldnt be any after. simple cause and effect i guess. there can have been no start of time, because a start indicates something previously, and even a lack of measureable time would still be a period of time.
You have to look at things from two standpoints: One is the way things are now, and the other is the way things were at the moment of the BB. As you have mentioned, and as I mentioned, too, somewhere in my ramblings, everything in our universe was created from a dense point of energy…. or that’s the theory. Right after the BB, we still didn’t have any matter. It took a brief period for the energy to cool enough to create the matter we have now.
How could we have a point of energy before the BB? I don’t know. I wasn’t around back then. Which is and always will be the same puzzlement as the concept of a God. Where did God come from? If he/she/it is a real thing, even energy counts as real, then were did that energy or matter come from to create him/her/it?
Our feeble minds cannot begin to grasp any of this. How can something just be? How could a God just exist? If not a God, how could a point of energy just exist? The great physicist, Stephen Hawking, had a circular/loop theory about how we could exist without a beginning…I don’t know.
I don’t know if we can measure time before the BB, because we have no measuring instruments, and the rules could be completely different. There would be no concept of time, as we know it. Nobody knows the answers to these deep questions, because we can only go back so far and that’s it. I believe, theoretically, we can go back to 10 times the 46th power of a second before the BB; A tiny fraction of a second. I forget now, been too long since I’ve read about it, but there has to be all sorts of things on the net about it. After that, the laws of physics break down.
How could we have a point of energy before the BB? I don’t know. I wasn’t around back then. Which is and always will be the same puzzlement as the concept of a God. Where did God come from? If he/she/it is a real thing, even energy counts as real, then were did that energy or matter come from to create him/her/it?
Our feeble minds cannot begin to grasp any of this. How can something just be? How could a God just exist? If not a God, how could a point of energy just exist? The great physicist, Stephen Hawking, had a circular/loop theory about how we could exist without a beginning…I don’t know.
I don’t know if we can measure time before the BB, because we have no measuring instruments, and the rules could be completely different. There would be no concept of time, as we know it. Nobody knows the answers to these deep questions, because we can only go back so far and that’s it. I believe, theoretically, we can go back to 10 times the 46th power of a second before the BB; A tiny fraction of a second. I forget now, been too long since I’ve read about it, but there has to be all sorts of things on the net about it. After that, the laws of physics break down.
phew
BACKGROUND
It is impossible for a human to travel very near a high gravity star which has a mass like that of the Sun. If, somehow, a person could survive the extremely harmful radiation that would be emitted on or near these objects, the high gravity itself would likely pose insurmountable problems. The person could not stand casually on the surface of such a star because the high surface gravity would tend to flatten them. (Lying down wouldn't help.) Were a person to orbit the star in a spaceship, however, the immense gravitational field would be overcome by a large outward centrifugal acceleration.[1] The problem in this case, however, is the extreme change in gravity between the head and toe of the person, the extreme tidal pull, would surely prove much more than annoying for any human[2].
Nevertheless it is informative and interesting to wonder what it would look like to visit such a high gravity environment. Significant speculations on this include popular science fiction stories such as those by Forward [3] and Niven [4]. A discussion (with cartoon drawings) of a trip to a black hole appears in Kaufmann's book ``The Cosmic Frontiers of General Relativity" [2]. A description of what hot spots on a high gravity neutron star would look like to an observer far away is given by Ftaclas, Kearney, and Pechenick, [5]. Other descriptions include what a typical neutron star would look like to a distant observer including a computer drawn wire mesh diagram [6], a description of the sky as seen from the vicinity of a black hole [7-9], a description of the image of a thin accretion disk around a black hole [10], a description of how the observer would see self-images near a black hole [11], and a short computer animated movie simulating a trip around a black hole while facing the constellation Orion by Palmer and Unruh [12]. In general, however, the professional science literature has focused mainly on mathematical detail rather than observable image distortions.
In this paper the visual aspects of a journey to several different types of high gravity stars will be discussed in some detail, along with computer generated illustrations highlighting the perceived visual distortions. The three types of stars that will be discussed are a) a "normal" neutron star having relatively weak surface gravity, b) a black hole, and c) an "ultracompact" neutron star [13] having relatively strong surface gravity. Here the speed of the traveler will always be considered small when compared to the local speed of light, so special relativistic effects will be ignored.
The paper is structured as follows: Section II discusses the physical principles and mathematics necessary to describe the perceived visual distortions. In Section III the types of visual distortions will be discussed generally. Section IV then proceeds to take the reader on a fantasy mission to these high gravity environments and describes what visual distortion effects the viewer would see. In Section V comments are made.
The visual distortion that will be described here would be caused by gravitation in the Schwarzschild metric [14]. Einstein's general relativity [15] is not the only gravitational theory that admits the Schwarzschild metric as an exterior solution for a spherically symmetric, non-rotating gravitational field, but it is the preferred theory, and the theory that will be assumed implicitly here. The Schwarzschild metric is:
Here ds is a metric measure of coordinate distance r, coordinate time t and coordinate angles theta and phi. The term R_S, the Schwarzschild radius, refers to the radius of a black hole event horizon, and c refers to the local speed of light. R_S is directly proportional to the mass that creates the metric through R_S = 2GM/c^2, where G is the gravitational constant and M is the mass interior to r.
For a photon, ds^2 = 0. Combining this with the conservation of angular momentum allows one to express the deflection angle phi of a photon moving in a gravitational field [16] as
where b is a constant over the trajectory of the photon path, corresponding to a linear projected impact parameter of a photon at infinity for a photon that escapes. This impact parameter can be visualized by assuming that when the photon is far from the gravitating object it travels in a straight line; the impact parameter is the distance between the closest approach of the continuation of this straight line and the center of the gravitating object. Note that Delta phi is not the extra angle deflected by the lens but the total change in the phi angle between the observer and the source, emitted at radial coordinate r_emitted and observed at radial coordinate r_observed. This angle is measured with the lens at the vertex, and includes gravitational deflection. Therefore, for example, a source seen by an observer just over the limb of a lens which has only a small mass, and hence a negligible effect of the trajectory of the photon, has a Delta phi near pi.
An important radius is found from Eq. (2) when Delta phi diverges to infinity. Here a photon will circle the massive star at the photon sphere. The exact location of the photon sphere is R_P = 1.5 R_S. Note that a "normal" neutron star with a relatively weak external gravitational field does not have a photon sphere. Were it somewhat more compact, it would have a photon sphere, and were it even more compact, it would have an event horizon and be called a black hole. For black holes and the "ultracompact" neutron stars considered below, however, these circular photon orbits can exist.
Photons circling at the photon sphere are not in a stable orbit [16] - any small perturbation will cause them to spiral either in or out. Photons emitted from infinity with impact parameters slightly greater than R_B = 3^(1.5) R_S / 2 will spiral around the compact star near the photon sphere and then spiral out. Photons emitted from infinity with impact parameters slightly less than R_B will spiral around near the photon sphere and then spiral in, eventually colliding with the neutron star surface or falling into the black hole. It is also possible for a photon to be emitted from a ultracompact neutron star surface, orbit near the photon sphere, and then spiral back in again impacting the surface. These describe, in general all of the distinct cases of photon orbit near an ultracompact neutron star. All shorter photon trajectories will lie on one of these paths.
Stated differently, the three cases of photon orbits near a gravitating body can be classified as: "always outside the photon sphere," "crossing the photon sphere," and "always inside the photon sphere." The first is the case of a photon passing the neutron star or black hole, reaching a critical radius R_c, and then escaping again toward infinity. In this case the photon does not reach or cross the photon sphere. Its distance from the star decreases monotonically until R_c, and then increases monotonically thereafter. The second case is that of a photon continuing to come toward the neutron star (or black hole) until it impacts the surface (or falls through the event horizon). Here its distance decreases monotonically. The third case is that of a photon emitted from the surface of a strong gravity neutron star, reaching a critical radius R_c, and then falling back down and again impacting the neutron star surface. This critical radius is given by the cubic equation solution [9]
where n = 0 is for the first case and n = 2 is for the third case.
Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a "redshifting", as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting. The observed energy E_observed at radius r_observed of a photon emitted at radius r_emitted with energy E_emitted is [7]
Note that the magnitude of the redshifting (blueshifting) effect is not a function of the emitted angle or the received angle of the photon - it depends only on how far radially the photon had to climb out of (fall into) the potential well. Also note that the power received from a continuously emitting source would have an additional factor of [(1 - R_S/r_emitted) / (1 - R_S/r_observed)]^(1/2) caused by the relative differences in the perceived rate of the number of photons emitted per unit time.
The effect a gravitational field would have on the actual perceived color of an object is more complex, however, as it depends on the distribution of photons emitted from the source at different energies relative to the sensitivity of the observer to measuring photons of different energies. For example, an object that would be described as green might be very bright in the ultra-violet - but this would not normally be perceived, as people cannot see the ultra-violet. Were this object put in a strong gravitational field and viewed from far away, so that the photons would be significantly redshifted, the strong ultra-violet emission could be shifted into violet emission and the object would look more blue, even though its light has been redshifted. This is an exceptional case, however, and redshifted objects may indeed appear more red.
VISUAL DISTORTION EFFECTS IN A HIGH GRAVITY ENVIRONMENT
A. Multiple images and amplification
A gravitational field may cause a single point source to appear with multiple images. For a spherical field all of these images will occur in the plane defined by the observer position, the center of the lens, and the source. These images cannot appear in other locations because this would break the principle of conservation of angular momentum along the photon orbits. Therefore all images of a single point source will appear on a single great circle on the observer's sky.
A gravitational field may cause an extended source to appear not only multiply imaged but also greatly distorted. There is at least one feature that each of the images will maintain, however, that is the same as the original source: red- or blueshift corrected surface brightness. Any radiative process preserves the specific intensity along the beam. [17] When gravity is involved, power along the beam is not conserved, it grows or shrinks in accordance with the red- or blueshift. What is conserved is the "corrected surface brightness" B_c = B_r / (1 - R_S/r), where B_r is the measured surface brightness at r.
For example, if an observer originally saw an unlensed circular source with constant surface brightness, a gravitational field could cause the observer to see multiple, elongated, images. Each image would have, however, the same corrected surface brightness (B_c) as the original unlensed source.
The net flux that reaches the observer from any single image of the source can be either more or less than the original unlensed flux of the source. Each image will undergo an amplification A, with A not constrained to be greater than unity. [9] This means that when considered together, the images of a source seen near a large gravitational field can have more or less flux than the same source seen without the intervening gravitational field. Essentially, there are two types of amplifications a source can be seen to undergo: time distortion induced amplifications A_time, related directly to the slowing of time in a gravitational field that causes photons to change both their energy (red- or blueshift) and the perceived arrival rate (and hence the source's perceived power integrated over all wavelengths), and amplifications in the apparent angular size of the source, A_angular. The total amplification will be designated A_total = A_time * A_angular. In the convention used here, all amplifications will be greater than zero.
Time induced amplifications result when the observer is at a different r from the lens center than the source. When considering only time induced amplifications, the total bolometric (incorporating all wavelengths) power received will then be changed by an amount A_time = (1-R_S/r_emitted)/(1-R_S/r_observed).
For the sources near perfect lens - observer alignment, the angular amplification effects typically dominate over time induced amplification effects. Angular amplifications can be computed from the deflection angle Eq. (2). If a large change in angular position on the observer's sky corresponds with a small change in the angular position at the (unlensed) source location, then the source will appear to be angularly elongated and hence amplified. Similarly a source can be angularly deamplified, but this will be referred to as an angular amplification of less than unity. Angular amplification effects should be computed on the spherical sky of the observer, and so would be given by [8]
Here beta represents the angular distance between the lens and the source on the observer's sky in the absence of the gravitational field of the lens, while alpha represents this distance in the presence of the gravitational field and light deflection. The change in this angular distance, d alpha/d beta can be found by application of Eq. (2). When bending angles due to gravitational effects are small, this reduces to the amplification formulae given by Refsdal [18] and Liebes. [19]
The net angular amplification of all the images of a single source can also be either more or less than the original unlensed flux of the source. [7, 20] This is because the gravitational field does not change the fact that the observer still observes the same total angular area as before: 4 pi steradians. Therefore if the apparent angular area from some sources is greater than without the gravitational field, then there must be other sources with apparent angular area which is lower, to compensate. In practice, only relatively few images from sources that are near the observer - lens line will have angular amplifications very large (A_angular >> 1), while the rest of the sources in the sky will be slightly deamplified (A_angular < ~ 1).
The total flux received by an observer from all the sky can again be either more or less than the original unlensed flux of all the sky. [20] A gravitational field does not create photons - it just redistributes and (red- or) blueshifts them. The observed angular redistribution and the relative time distortions, however, now act in opposite directions. For the background sky, A_angular < 1, because now its angular area, which used to occupy the observer's entire field of view (4 pi steradians) in the absence of gravity, is now less by the amount of the angular size of the photon sphere of the lens. However, the photons from the sky, because of the blueshifting, are relatively more energetic and arriving relatively more often, so that A_time > 1. [7]
In other words, the background sky takes up less of the observer's sky, but the observer receives more photons per unit area, and each photon is of higher energy. Do these effects exactly cancel? No. It turns out that all observers will measure A_total > 1, with the closer the oberver the greater is A_total.
B. Einstein rings
An important observational aspect of visual distortions in a high gravity environment that is discussed more usually in the gravitational lensing literature than in the introductory gravitation texts is called an Einstein ring. [18-19, 21] Before it was shown that all images must occur in the plane defined by the observer's position, the center of the lens, and the point source. But what if these are all collinear? No plane is then defined. In this case the image of the point source would appear to the observer as an infinitesimally thin ring. This is an Einstein ring. As will be explained, numerous Einstein rings may appear simultaneously, however, and they are also important as invisible dividing lines between sets of images, [7] even when no source is distorted into a ring.
It is not generally appreciated that there can be an infinite number of Einstein rings. In fact, there can be an infinite number of Einstein rings for each set of collinear observer, lens, and source points. The only Einstein ring currently discussed in the literature is the most prominent one that occurs at precise observer - lens - source alignment, where Delta phi = pi. Here light emitted at a specific angle from the source would be slightly deflected by the gravitational field of the lens to reach the observer. Were the source light emitted at a different angle the lens would either not be able to bend the light enough to reach the observer or too much. Since the exact observer, lens, source alignment is symmetric about the line connecting them, this source would be seen as an annular ring. This ring will be referred to as the first Einstein ring. (Later the term Einstein ring will be even additionally labelled by the relative radius of the source.)
Other Einstein rings can be seen angularly closer to the center of the lens. Photons from the third Einstein ring (the second Einstein ring will be defined two paragraphs below) have fully circled the lens once near the photon sphere before coming to the observer. In fact, the path of these photons crosses itself. It is possible for photons to orbit the lens an arbitrarily large number of times before coming to the observer, and each of these orbits corresponds to an Einstein ring. Therefore there are innumerable Einstein rings for this specific observer - lens - source configuration. Each Einstein ring is seen successively closer to the apparent photon sphere position. The more times the photon must circle the neutron star or black hole before reaching the observer, the more precise the direction of its emission must have been emitted to attain this trajectory, the less likely any photon will take this trajectory, the "dimmer" the Einstein ring. For this reason the higher order Einstein rings will usually carry little light when compared to the lower order Einstein rings. In fact, the relative brightness of each Einstein ring decreases exponentially. [10]
The first Einstein ring can be seen not only in a high gravity environment, but also in a low gravity environment quite a distance from much larger objects, such as normal stars, galaxies, and clusters of galaxies. In fact, complete first Einstein rings have actually been seen for radio galaxies. [22] A good review of extragalactic measurements of gravitational lens effects is given by Blandford and Narayan. [23]
Another set of Einstein rings is observable when the observer and source are on the same side of the lens. Then, for compact sources such as an ultracompact neutron star, light from behind the observer is able to make a "U-turn" around the neutron star and come back to be visible to the observer. The Einstein ring seen from these light trajectories will be called the second Einstein ring, since it is seen between the first and third Einstein rings, and is brighter than the third but dimmer than the first. The fourth Einstein ring in this set is created when light does a "U-turn" near the photon sphere of the lens, then goes all the way around the lens again near the photon sphere, and finally comes to the observer. Note that there is a critical minimum (or maximum for observers inside the photon sphere) distance for the photon just like in the case of slight deflection, that is given by Eq. (3). There are also an infinite number of higher order Einstein rings of this type. As before, however, these Einstein rings carry relatively little power when compared to the lower order Einstein rings.
It is convenient to also define the zeroth Einstein ring, where light from a source located on the line from the lens through the observer comes directly undeflected to the observer along a radial line (Delta phi = 0). This Einstein "ring" is actually a single point on the observer's sky. It differs from the other Einstein rings in that its angular amplification (of a collinear point source) is not formally divergent.
Note that a single source located precisely on the opposite side of the lens from the observer would create only the first, third, fifth, etc. (i.e. odd numbered) Einstein rings. A single source located on the same side of the lens as the observer would create the zeroth, second, fourth, etc. (i.e. even numbered) Einstein rings.
In general, the position of each set of Einstein rings will be different for each specific source radius from the lens, relative to the observer position. For example, a point source at infinity directly behind the lens from the observer would create a complete set of odd numbered Einstein rings. A point source located a small, finite distance from the lens (but still directly behind the lens) would create a different set of odd numbered Einstein rings. Each set of Einstein rings can thus be labeled by the location of the source sphere. Sources at infinity will be referred to "sky" Einstein rings. For sources on the surface of the lens, the term "surface" Einstein rings will be used. In general, the convention will be taken of labelling each Einstein ring by the name or radius of the source sphere.
Mathematically, an Einstein ring will always occur when the total deflection angle due to gravitation Delta phi (Eq. 2) is equal to any integer multiple of pi radians. [7] Note that the Einstein rings are theoretical constructs and would only be visible were a source placed precisely on the observer-lens line, which for any small source is unlikely.
If the angular radius of an opaque lens is larger than the angular radius of the first Einstein ring for the source, then this ring will exist. If the radius of the lens is smaller than the radius of the first Einstein ring but larger than the other Einstein rings, then only the first Einstein ring will exist. If the radius of the lens is small enough so that the lens exhibits a photon sphere, however, an infinite number of Einstein rings exist. This is because a subsequent Einstein ring exists for each revolution of the lens a photon orbit can take, and theoretically, since all of these orbits are contained completely above the photon sphere, it can take an infinite number of them.
It should be noted that the existence of an Einstein ring may depend on the relative positions of the lens, observer, and source, while the existence of the photon sphere or event horizon does not depend on these relative positions. It is possible for the first sky Einstein ring to exist for a given observer looking toward a neutron star lens, but as the observer moves closer to the neutron star the angular size of the surface becomes larger than the angular size of this Einstein ring. For black holes and neutron star's compact enough to have a photon sphere, though, the photon sphere is a real entity - photons do circle there - whether or not an observer is there to see them.
C. Complete sky and surface visibility
A complete image of the sky is always contained between each two "sky" Einstein rings. [7] Likewise a complete image of the neutron star is always contained between each two "surface" Einstein rings. In general, a single complete image of all the sources on a sphere centered on the lens is visible between each two consecutive Einstein rings of that sphere.
If the radius of the lens is small enough so that the lens exhibits a photon sphere, an infinite number of images can be seen of the source, no matter its location. One image of the source comes to the observer relatively undeflected. This image is between the zeroth and first Einstein rings and will be referred to as the primary image. A second image comes around the opposite limb of the lens from the first image, and therefore will appear to the observer 180 degrees around the face of the lens from the first image. This secondary image will always be located between the first and second Einstein rings. A third image comes around the same limb as the first image and is seen even closer to the apparent position of the photon sphere. This image has circled the neutron star or black hole fully once before reaching the observer, and its location is always between the second and third Einstein rings. The photon path for this image (and all higher order images) crosses itself. A fourth image occurs closer to but outside of the same limb as the second image, but has fully circled the lens once in the opposite direction. There is a subsequent image for each revolution of the lens a photon orbits takes, and theoretically it can take an infinite number of them. In practice, these multi-revolution images have little power and would be vanishingly hard to see. [9]
Each set of images contained between successive Einstein rings is converted into "mirror writing" with respect to the images between the previous two Einstein rings. For example, if the source was a book, then the book would be visible with relatively minor distortions in its primary image - between the zeroth and first book Einstein rings. For the secondary image, between the first and second book Einstein rings, the book would appear in mirror writing, but right side up. The mapping of the entire sphere onto the annular ring between the two book Einstein rings would also cause prominent distortions. The third image of the book, between the second and third book Einstein rings, would appear in normal writing again (neither in mirror writing nor inverted), but even more distorted because of the decreased relative angular area between these two book Einstein rings. A discussion of the parity of lensed images for the brightest two images of the point lens (considered here) as well as other gravitational lens types, can be found in Blandford and Kochanek. [24]
Therefore, for a compact enough neutron star, one can see the whole neutron star surface. [5] An observer can see the complete surface of a lens (exactly once) when the first surface Einstein ring is the same angular size as the surface of the lens. (A derivation of the angular size of a sphere of mass M, radius R_* at distance D is given in the Appendix.) When the second surface Einstein ring has equal angular size to the apparent angular size of the lens surface, two complete images of the lens surface are visible.
Any lens which has a first surface Einstein ring is completely incapable of blocking light from any source. These objects cannot "eclipse" anything. This is why a neutron star in a well separated binary system can never block the light of its binary companion.
Less stringently, any lens with a first sky Einstein ring is incapable of blocking light from the background sky. Almost all stars in our galaxy are thus incapable of blocking light from random superpositions of background objects. For example, no supernovae in other galaxies are missed because they are "eclipsed" by a random superposition of a foreground star in the Milky Way Galaxy. Were such a chance superposition to occur (it is very unlikely), the supernova would be greatly amplified by the gravitational field of the intervening star rather than diminished by an "eclipse" effect. With respect to distant sources, these stars are easily compact enough to show a first Einstein ring to a distant observer, and are therefore incapable of blocking the source's light.
Every star in existence, besides the Sun but including even the nearest stars, has a first sky Einstein ring with respect to an earth bound observer. The small angular size of this Einstein ring is below optical resolution, but not below the angular resolution of many radio observations. The gravity of these normal stars is strong enough to bend the background light around them and cause distant sources to be visible to the observer. Almost none of the nearby stars, however, would show a second sky Einstein ring, unless they were a neutron star or black hole. Were the star compact enough to have a photon sphere surrounding it, then, theoretically, an infinite number of sky Einstein rings (and hence sky images) would be visible.
D. "Self" Einstein rings: Where to see yourself
A very interesting set of Einstein rings are the ``self" Einstein rings, where observers can see themselves. The most well known of these can be seen when the observer is located at the photon sphere. There observers can simply look along the photon sphere, where light travels in a circle, and see the backs of their heads! [11,25] All observers in the presence of a sufficiently compact lens, however, can see themselves. Here, light can leave the observer, travel around the lens and return to the observer to be viewed. Observers would see themselves as a series of Einstein rings. The more times light can circle the lens and return to the observer, the more "self" images the observer can see. For a lens compact enough to have a photon sphere, observers can, theoretically, see themselves in every self Einstein ring: an infinite number of times.
Amusingly, there is only a single case where observers can see only a single image of themselves - and this is the case that is well known [25] - when observers are at the photon sphere! Here all the self Einstein rings actually merge with the photon sphere to form a single observer image.
Observers who see themselves would be viewing themselves with high amplification. This is because the self images observers would see would be on or near Einstein rings - which carry the highest amplifications. Therefore gravity has become a powerful microscope! When at the photon sphere observers can microscopically view the backs of their heads, and when far away observers can microscopically view their own eyes. This is because the light that returns to the observer has left on a nearly radial trajectory - and the part of the observer most nearly radial is the observer's own eye. When close to and inside the photon sphere, observers can inspect annular rings on their heads (or spacecrafts).
IV. JOURNEY TO A HIGH GRAVITY STAR
A fairly detailed description of the distortion effects a space-traveler (or camera) would see on a visit to a high gravity star is now possible. The case that will be described first will be a trip to a "normal" neutron star: one with a currently popular equation for the interior structure of the star. This star is not dense enough to have an event horizon or photon sphere.
The second case that will be described is that of visiting a black hole. This case is more complex in that many bound and unbound photon orbits exist near the black hole. There is, however, a somewhat simpler aspect to describing this case than the previous one in that one does not have to track surface feature distortions for a black hole.
The third and last case that will be described is that of visiting a ultracompact neutron star - one with an extreme equation for its interior structure that allows a mean density so high the star has a photon sphere. This is the most complicated case of all to describe as it involves all three types of photon orbits described above as well as requiring a description of both the sky and surface feature distortions.
To more clearly delineate what the viewer would see, a set of computer generated figures were created that document the distortion effects in terms of familiar icons. In these illustrations, the sky in the background behind the high gravity star was taken to be the night sky as viewed from present-day earth. More specifically, the background sky is taken from the Bright Star Catalogue, [26] allowing all stellar images as dim as 5th magnitude to be seen, and stellar images as dim as 7th magnitude may be amplified into visibility. In the two cases of neutron stars, a map of the earth was projected onto the surfaces of the stars and allowed to distort. These figures are, in many aspects, fully general relativistically correct. The resolution of the figures is about 3 arcminutes (0.05 degrees).
Stellar image brightnesses are shown by the area the stellar image takes on the plots: the area is directly proportional to the flux the observer would receive from the image. It was impossible to change the pixel brightness, so many of the single pixel images would actually be seen dimmer than shown in the figures. Stellar images were allowed to get brighter or dimmer by angular amplification effects, but time induced amplification effects have been suppressed.
Note that for A_angular > 1 the stellar image flux would actually be seen as an increase in angular area of the image, so that the amplified angular area of the stellar images in the computer generated plots are, in this sense, realistic. However, the distortions in the amplified images would not be readily observable, as these background images would be unresolved by the viewer and hence indistinguishable from point sources. A small amplification would not cause the image to be resolved. Stellar images will therefore always be depicted as circles, even when they undergo angular amplification, as these convey best the idea of an unresolved point sources.
Only the two brightest images of all sources were tracked by the computer programs used. All stars originally 5th magnitude or brighter are plotted as secondary images, no matter their magnitude after gravitational distortion. Stars originally 5th magnitude are only plotted as primary images, however, if their final post-lensed magnitude was 5 or brighter. Higher order images undergoing larger angular amplification could potentially be visible but one would need significantly better angular resolution so see them (the only exception to this will be Fig. 2p), so they will be suppressed. An angular amplification limit of a factor of 100 was placed on all images for plotting purposes.
The hypothetical "camera" used in the simulations is somewhat fanciful but has several defining characteristics. First of all the camera is asymptotically small so that no general relativistic light bending effects are important over the length of the camera. The camera's field of view is 90 degrees across the middle of the picture. Lastly, the illustrations that follow, produced by the "camera," have been "flat-fielded" so that angular area on the spherical sky is directly proportional to spatial area on the flat page.
And always i must please you monkeys with proof of research
Frontiers of General Relativity (Little, Brown, and Company, Boston, 1977), pp. 120-150.
R. L. Forward, Dragon's Egg (Ballantine, New York, 1980).
L. Niven, Neutron Star (Ballantine, New York, 1968), p. 9.
C. Ftaclas, M. W. Kearney, and Pechenick, K. R., "Hot Spots on Neutron Stars. II - The Observer's Sky," Astrophys. J. 300, 203-208 (1986).
H.-P. Nollert, H. Ruder, H. Herold, and U. Kraus, "The Relativistic `Looks' of a Neutron Star," Astron. Astrophys. 208, 153-156 (1988).
C. T. Cunningham, "Optical Appearance of Distant Observers near and Inside a Schwarzschild Black Hole," Phys. Rev. D. 12, 323-328 (1975).
J. Schastok, M. Soffel, H. Ruder, and M. Schneider, "Stellar Sky as Seen From the Vicinity of a Black Hole," Am. J. Phys. 55, 336-341 (1987).
H. C. Ohanian, "The Black Hole as a Gravitational `Lens'," Am. J. Phys. 55, 428-432 (1987).
J.-P. Luminet, "Image of a Spherical Black Hole with Thin Accretion Disk," Astron. Astrophys. 75, 228-235 (1979).
W. M. Stuckey, "The Schwarzschild Black Hole as a Gravitational Mirror," Am. J. Phys., submitted (1992).
L. Palmer and W. Unruh, shown at a Texas Symposium on Relativistic Astrophysics in the late 1970s.
B. R. Iyer, C. V. Vishveshwara, S. V. Dhurandhar, "Ultracompact (R less than 3 M) Objects in General Relativity," Class. Quant. Grav. 2, 219-228 (1985).
K. Schwarzschild, "Ueber das Gravitationalsfeld einer Massenpunktes nach der Einsteinschen Theorie," Sitzunsgsber. dtsch. Akad. Wiss. Berlin, 189-196 (1916).
A. Einstein, "Die Grundlage der allgemeinen Relativitatstheorie," Ann. Phys. 49, 769-822 (1916).
and here you have it.
BACKGROUND
It is impossible for a human to travel very near a high gravity star which has a mass like that of the Sun. If, somehow, a person could survive the extremely harmful radiation that would be emitted on or near these objects, the high gravity itself would likely pose insurmountable problems. The person could not stand casually on the surface of such a star because the high surface gravity would tend to flatten them. (Lying down wouldn't help.) Were a person to orbit the star in a spaceship, however, the immense gravitational field would be overcome by a large outward centrifugal acceleration.[1] The problem in this case, however, is the extreme change in gravity between the head and toe of the person, the extreme tidal pull, would surely prove much more than annoying for any human[2].
Nevertheless it is informative and interesting to wonder what it would look like to visit such a high gravity environment. Significant speculations on this include popular science fiction stories such as those by Forward [3] and Niven [4]. A discussion (with cartoon drawings) of a trip to a black hole appears in Kaufmann's book ``The Cosmic Frontiers of General Relativity" [2]. A description of what hot spots on a high gravity neutron star would look like to an observer far away is given by Ftaclas, Kearney, and Pechenick, [5]. Other descriptions include what a typical neutron star would look like to a distant observer including a computer drawn wire mesh diagram [6], a description of the sky as seen from the vicinity of a black hole [7-9], a description of the image of a thin accretion disk around a black hole [10], a description of how the observer would see self-images near a black hole [11], and a short computer animated movie simulating a trip around a black hole while facing the constellation Orion by Palmer and Unruh [12]. In general, however, the professional science literature has focused mainly on mathematical detail rather than observable image distortions.
In this paper the visual aspects of a journey to several different types of high gravity stars will be discussed in some detail, along with computer generated illustrations highlighting the perceived visual distortions. The three types of stars that will be discussed are a) a "normal" neutron star having relatively weak surface gravity, b) a black hole, and c) an "ultracompact" neutron star [13] having relatively strong surface gravity. Here the speed of the traveler will always be considered small when compared to the local speed of light, so special relativistic effects will be ignored.
The paper is structured as follows: Section II discusses the physical principles and mathematics necessary to describe the perceived visual distortions. In Section III the types of visual distortions will be discussed generally. Section IV then proceeds to take the reader on a fantasy mission to these high gravity environments and describes what visual distortion effects the viewer would see. In Section V comments are made.
The visual distortion that will be described here would be caused by gravitation in the Schwarzschild metric [14]. Einstein's general relativity [15] is not the only gravitational theory that admits the Schwarzschild metric as an exterior solution for a spherically symmetric, non-rotating gravitational field, but it is the preferred theory, and the theory that will be assumed implicitly here. The Schwarzschild metric is:
Here ds is a metric measure of coordinate distance r, coordinate time t and coordinate angles theta and phi. The term R_S, the Schwarzschild radius, refers to the radius of a black hole event horizon, and c refers to the local speed of light. R_S is directly proportional to the mass that creates the metric through R_S = 2GM/c^2, where G is the gravitational constant and M is the mass interior to r.
For a photon, ds^2 = 0. Combining this with the conservation of angular momentum allows one to express the deflection angle phi of a photon moving in a gravitational field [16] as
where b is a constant over the trajectory of the photon path, corresponding to a linear projected impact parameter of a photon at infinity for a photon that escapes. This impact parameter can be visualized by assuming that when the photon is far from the gravitating object it travels in a straight line; the impact parameter is the distance between the closest approach of the continuation of this straight line and the center of the gravitating object. Note that Delta phi is not the extra angle deflected by the lens but the total change in the phi angle between the observer and the source, emitted at radial coordinate r_emitted and observed at radial coordinate r_observed. This angle is measured with the lens at the vertex, and includes gravitational deflection. Therefore, for example, a source seen by an observer just over the limb of a lens which has only a small mass, and hence a negligible effect of the trajectory of the photon, has a Delta phi near pi.
An important radius is found from Eq. (2) when Delta phi diverges to infinity. Here a photon will circle the massive star at the photon sphere. The exact location of the photon sphere is R_P = 1.5 R_S. Note that a "normal" neutron star with a relatively weak external gravitational field does not have a photon sphere. Were it somewhat more compact, it would have a photon sphere, and were it even more compact, it would have an event horizon and be called a black hole. For black holes and the "ultracompact" neutron stars considered below, however, these circular photon orbits can exist.
Photons circling at the photon sphere are not in a stable orbit [16] - any small perturbation will cause them to spiral either in or out. Photons emitted from infinity with impact parameters slightly greater than R_B = 3^(1.5) R_S / 2 will spiral around the compact star near the photon sphere and then spiral out. Photons emitted from infinity with impact parameters slightly less than R_B will spiral around near the photon sphere and then spiral in, eventually colliding with the neutron star surface or falling into the black hole. It is also possible for a photon to be emitted from a ultracompact neutron star surface, orbit near the photon sphere, and then spiral back in again impacting the surface. These describe, in general all of the distinct cases of photon orbit near an ultracompact neutron star. All shorter photon trajectories will lie on one of these paths.
Stated differently, the three cases of photon orbits near a gravitating body can be classified as: "always outside the photon sphere," "crossing the photon sphere," and "always inside the photon sphere." The first is the case of a photon passing the neutron star or black hole, reaching a critical radius R_c, and then escaping again toward infinity. In this case the photon does not reach or cross the photon sphere. Its distance from the star decreases monotonically until R_c, and then increases monotonically thereafter. The second case is that of a photon continuing to come toward the neutron star (or black hole) until it impacts the surface (or falls through the event horizon). Here its distance decreases monotonically. The third case is that of a photon emitted from the surface of a strong gravity neutron star, reaching a critical radius R_c, and then falling back down and again impacting the neutron star surface. This critical radius is given by the cubic equation solution [9]
where n = 0 is for the first case and n = 2 is for the third case.
Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a "redshifting", as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting. The observed energy E_observed at radius r_observed of a photon emitted at radius r_emitted with energy E_emitted is [7]
Note that the magnitude of the redshifting (blueshifting) effect is not a function of the emitted angle or the received angle of the photon - it depends only on how far radially the photon had to climb out of (fall into) the potential well. Also note that the power received from a continuously emitting source would have an additional factor of [(1 - R_S/r_emitted) / (1 - R_S/r_observed)]^(1/2) caused by the relative differences in the perceived rate of the number of photons emitted per unit time.
The effect a gravitational field would have on the actual perceived color of an object is more complex, however, as it depends on the distribution of photons emitted from the source at different energies relative to the sensitivity of the observer to measuring photons of different energies. For example, an object that would be described as green might be very bright in the ultra-violet - but this would not normally be perceived, as people cannot see the ultra-violet. Were this object put in a strong gravitational field and viewed from far away, so that the photons would be significantly redshifted, the strong ultra-violet emission could be shifted into violet emission and the object would look more blue, even though its light has been redshifted. This is an exceptional case, however, and redshifted objects may indeed appear more red.
VISUAL DISTORTION EFFECTS IN A HIGH GRAVITY ENVIRONMENT
A. Multiple images and amplification
A gravitational field may cause a single point source to appear with multiple images. For a spherical field all of these images will occur in the plane defined by the observer position, the center of the lens, and the source. These images cannot appear in other locations because this would break the principle of conservation of angular momentum along the photon orbits. Therefore all images of a single point source will appear on a single great circle on the observer's sky.
A gravitational field may cause an extended source to appear not only multiply imaged but also greatly distorted. There is at least one feature that each of the images will maintain, however, that is the same as the original source: red- or blueshift corrected surface brightness. Any radiative process preserves the specific intensity along the beam. [17] When gravity is involved, power along the beam is not conserved, it grows or shrinks in accordance with the red- or blueshift. What is conserved is the "corrected surface brightness" B_c = B_r / (1 - R_S/r), where B_r is the measured surface brightness at r.
For example, if an observer originally saw an unlensed circular source with constant surface brightness, a gravitational field could cause the observer to see multiple, elongated, images. Each image would have, however, the same corrected surface brightness (B_c) as the original unlensed source.
The net flux that reaches the observer from any single image of the source can be either more or less than the original unlensed flux of the source. Each image will undergo an amplification A, with A not constrained to be greater than unity. [9] This means that when considered together, the images of a source seen near a large gravitational field can have more or less flux than the same source seen without the intervening gravitational field. Essentially, there are two types of amplifications a source can be seen to undergo: time distortion induced amplifications A_time, related directly to the slowing of time in a gravitational field that causes photons to change both their energy (red- or blueshift) and the perceived arrival rate (and hence the source's perceived power integrated over all wavelengths), and amplifications in the apparent angular size of the source, A_angular. The total amplification will be designated A_total = A_time * A_angular. In the convention used here, all amplifications will be greater than zero.
Time induced amplifications result when the observer is at a different r from the lens center than the source. When considering only time induced amplifications, the total bolometric (incorporating all wavelengths) power received will then be changed by an amount A_time = (1-R_S/r_emitted)/(1-R_S/r_observed).
For the sources near perfect lens - observer alignment, the angular amplification effects typically dominate over time induced amplification effects. Angular amplifications can be computed from the deflection angle Eq. (2). If a large change in angular position on the observer's sky corresponds with a small change in the angular position at the (unlensed) source location, then the source will appear to be angularly elongated and hence amplified. Similarly a source can be angularly deamplified, but this will be referred to as an angular amplification of less than unity. Angular amplification effects should be computed on the spherical sky of the observer, and so would be given by [8]
Here beta represents the angular distance between the lens and the source on the observer's sky in the absence of the gravitational field of the lens, while alpha represents this distance in the presence of the gravitational field and light deflection. The change in this angular distance, d alpha/d beta can be found by application of Eq. (2). When bending angles due to gravitational effects are small, this reduces to the amplification formulae given by Refsdal [18] and Liebes. [19]
The net angular amplification of all the images of a single source can also be either more or less than the original unlensed flux of the source. [7, 20] This is because the gravitational field does not change the fact that the observer still observes the same total angular area as before: 4 pi steradians. Therefore if the apparent angular area from some sources is greater than without the gravitational field, then there must be other sources with apparent angular area which is lower, to compensate. In practice, only relatively few images from sources that are near the observer - lens line will have angular amplifications very large (A_angular >> 1), while the rest of the sources in the sky will be slightly deamplified (A_angular < ~ 1).
The total flux received by an observer from all the sky can again be either more or less than the original unlensed flux of all the sky. [20] A gravitational field does not create photons - it just redistributes and (red- or) blueshifts them. The observed angular redistribution and the relative time distortions, however, now act in opposite directions. For the background sky, A_angular < 1, because now its angular area, which used to occupy the observer's entire field of view (4 pi steradians) in the absence of gravity, is now less by the amount of the angular size of the photon sphere of the lens. However, the photons from the sky, because of the blueshifting, are relatively more energetic and arriving relatively more often, so that A_time > 1. [7]
In other words, the background sky takes up less of the observer's sky, but the observer receives more photons per unit area, and each photon is of higher energy. Do these effects exactly cancel? No. It turns out that all observers will measure A_total > 1, with the closer the oberver the greater is A_total.
B. Einstein rings
An important observational aspect of visual distortions in a high gravity environment that is discussed more usually in the gravitational lensing literature than in the introductory gravitation texts is called an Einstein ring. [18-19, 21] Before it was shown that all images must occur in the plane defined by the observer's position, the center of the lens, and the point source. But what if these are all collinear? No plane is then defined. In this case the image of the point source would appear to the observer as an infinitesimally thin ring. This is an Einstein ring. As will be explained, numerous Einstein rings may appear simultaneously, however, and they are also important as invisible dividing lines between sets of images, [7] even when no source is distorted into a ring.
It is not generally appreciated that there can be an infinite number of Einstein rings. In fact, there can be an infinite number of Einstein rings for each set of collinear observer, lens, and source points. The only Einstein ring currently discussed in the literature is the most prominent one that occurs at precise observer - lens - source alignment, where Delta phi = pi. Here light emitted at a specific angle from the source would be slightly deflected by the gravitational field of the lens to reach the observer. Were the source light emitted at a different angle the lens would either not be able to bend the light enough to reach the observer or too much. Since the exact observer, lens, source alignment is symmetric about the line connecting them, this source would be seen as an annular ring. This ring will be referred to as the first Einstein ring. (Later the term Einstein ring will be even additionally labelled by the relative radius of the source.)
Other Einstein rings can be seen angularly closer to the center of the lens. Photons from the third Einstein ring (the second Einstein ring will be defined two paragraphs below) have fully circled the lens once near the photon sphere before coming to the observer. In fact, the path of these photons crosses itself. It is possible for photons to orbit the lens an arbitrarily large number of times before coming to the observer, and each of these orbits corresponds to an Einstein ring. Therefore there are innumerable Einstein rings for this specific observer - lens - source configuration. Each Einstein ring is seen successively closer to the apparent photon sphere position. The more times the photon must circle the neutron star or black hole before reaching the observer, the more precise the direction of its emission must have been emitted to attain this trajectory, the less likely any photon will take this trajectory, the "dimmer" the Einstein ring. For this reason the higher order Einstein rings will usually carry little light when compared to the lower order Einstein rings. In fact, the relative brightness of each Einstein ring decreases exponentially. [10]
The first Einstein ring can be seen not only in a high gravity environment, but also in a low gravity environment quite a distance from much larger objects, such as normal stars, galaxies, and clusters of galaxies. In fact, complete first Einstein rings have actually been seen for radio galaxies. [22] A good review of extragalactic measurements of gravitational lens effects is given by Blandford and Narayan. [23]
Another set of Einstein rings is observable when the observer and source are on the same side of the lens. Then, for compact sources such as an ultracompact neutron star, light from behind the observer is able to make a "U-turn" around the neutron star and come back to be visible to the observer. The Einstein ring seen from these light trajectories will be called the second Einstein ring, since it is seen between the first and third Einstein rings, and is brighter than the third but dimmer than the first. The fourth Einstein ring in this set is created when light does a "U-turn" near the photon sphere of the lens, then goes all the way around the lens again near the photon sphere, and finally comes to the observer. Note that there is a critical minimum (or maximum for observers inside the photon sphere) distance for the photon just like in the case of slight deflection, that is given by Eq. (3). There are also an infinite number of higher order Einstein rings of this type. As before, however, these Einstein rings carry relatively little power when compared to the lower order Einstein rings.
It is convenient to also define the zeroth Einstein ring, where light from a source located on the line from the lens through the observer comes directly undeflected to the observer along a radial line (Delta phi = 0). This Einstein "ring" is actually a single point on the observer's sky. It differs from the other Einstein rings in that its angular amplification (of a collinear point source) is not formally divergent.
Note that a single source located precisely on the opposite side of the lens from the observer would create only the first, third, fifth, etc. (i.e. odd numbered) Einstein rings. A single source located on the same side of the lens as the observer would create the zeroth, second, fourth, etc. (i.e. even numbered) Einstein rings.
In general, the position of each set of Einstein rings will be different for each specific source radius from the lens, relative to the observer position. For example, a point source at infinity directly behind the lens from the observer would create a complete set of odd numbered Einstein rings. A point source located a small, finite distance from the lens (but still directly behind the lens) would create a different set of odd numbered Einstein rings. Each set of Einstein rings can thus be labeled by the location of the source sphere. Sources at infinity will be referred to "sky" Einstein rings. For sources on the surface of the lens, the term "surface" Einstein rings will be used. In general, the convention will be taken of labelling each Einstein ring by the name or radius of the source sphere.
Mathematically, an Einstein ring will always occur when the total deflection angle due to gravitation Delta phi (Eq. 2) is equal to any integer multiple of pi radians. [7] Note that the Einstein rings are theoretical constructs and would only be visible were a source placed precisely on the observer-lens line, which for any small source is unlikely.
If the angular radius of an opaque lens is larger than the angular radius of the first Einstein ring for the source, then this ring will exist. If the radius of the lens is smaller than the radius of the first Einstein ring but larger than the other Einstein rings, then only the first Einstein ring will exist. If the radius of the lens is small enough so that the lens exhibits a photon sphere, however, an infinite number of Einstein rings exist. This is because a subsequent Einstein ring exists for each revolution of the lens a photon orbit can take, and theoretically, since all of these orbits are contained completely above the photon sphere, it can take an infinite number of them.
It should be noted that the existence of an Einstein ring may depend on the relative positions of the lens, observer, and source, while the existence of the photon sphere or event horizon does not depend on these relative positions. It is possible for the first sky Einstein ring to exist for a given observer looking toward a neutron star lens, but as the observer moves closer to the neutron star the angular size of the surface becomes larger than the angular size of this Einstein ring. For black holes and neutron star's compact enough to have a photon sphere, though, the photon sphere is a real entity - photons do circle there - whether or not an observer is there to see them.
C. Complete sky and surface visibility
A complete image of the sky is always contained between each two "sky" Einstein rings. [7] Likewise a complete image of the neutron star is always contained between each two "surface" Einstein rings. In general, a single complete image of all the sources on a sphere centered on the lens is visible between each two consecutive Einstein rings of that sphere.
If the radius of the lens is small enough so that the lens exhibits a photon sphere, an infinite number of images can be seen of the source, no matter its location. One image of the source comes to the observer relatively undeflected. This image is between the zeroth and first Einstein rings and will be referred to as the primary image. A second image comes around the opposite limb of the lens from the first image, and therefore will appear to the observer 180 degrees around the face of the lens from the first image. This secondary image will always be located between the first and second Einstein rings. A third image comes around the same limb as the first image and is seen even closer to the apparent position of the photon sphere. This image has circled the neutron star or black hole fully once before reaching the observer, and its location is always between the second and third Einstein rings. The photon path for this image (and all higher order images) crosses itself. A fourth image occurs closer to but outside of the same limb as the second image, but has fully circled the lens once in the opposite direction. There is a subsequent image for each revolution of the lens a photon orbits takes, and theoretically it can take an infinite number of them. In practice, these multi-revolution images have little power and would be vanishingly hard to see. [9]
Each set of images contained between successive Einstein rings is converted into "mirror writing" with respect to the images between the previous two Einstein rings. For example, if the source was a book, then the book would be visible with relatively minor distortions in its primary image - between the zeroth and first book Einstein rings. For the secondary image, between the first and second book Einstein rings, the book would appear in mirror writing, but right side up. The mapping of the entire sphere onto the annular ring between the two book Einstein rings would also cause prominent distortions. The third image of the book, between the second and third book Einstein rings, would appear in normal writing again (neither in mirror writing nor inverted), but even more distorted because of the decreased relative angular area between these two book Einstein rings. A discussion of the parity of lensed images for the brightest two images of the point lens (considered here) as well as other gravitational lens types, can be found in Blandford and Kochanek. [24]
Therefore, for a compact enough neutron star, one can see the whole neutron star surface. [5] An observer can see the complete surface of a lens (exactly once) when the first surface Einstein ring is the same angular size as the surface of the lens. (A derivation of the angular size of a sphere of mass M, radius R_* at distance D is given in the Appendix.) When the second surface Einstein ring has equal angular size to the apparent angular size of the lens surface, two complete images of the lens surface are visible.
Any lens which has a first surface Einstein ring is completely incapable of blocking light from any source. These objects cannot "eclipse" anything. This is why a neutron star in a well separated binary system can never block the light of its binary companion.
Less stringently, any lens with a first sky Einstein ring is incapable of blocking light from the background sky. Almost all stars in our galaxy are thus incapable of blocking light from random superpositions of background objects. For example, no supernovae in other galaxies are missed because they are "eclipsed" by a random superposition of a foreground star in the Milky Way Galaxy. Were such a chance superposition to occur (it is very unlikely), the supernova would be greatly amplified by the gravitational field of the intervening star rather than diminished by an "eclipse" effect. With respect to distant sources, these stars are easily compact enough to show a first Einstein ring to a distant observer, and are therefore incapable of blocking the source's light.
Every star in existence, besides the Sun but including even the nearest stars, has a first sky Einstein ring with respect to an earth bound observer. The small angular size of this Einstein ring is below optical resolution, but not below the angular resolution of many radio observations. The gravity of these normal stars is strong enough to bend the background light around them and cause distant sources to be visible to the observer. Almost none of the nearby stars, however, would show a second sky Einstein ring, unless they were a neutron star or black hole. Were the star compact enough to have a photon sphere surrounding it, then, theoretically, an infinite number of sky Einstein rings (and hence sky images) would be visible.
D. "Self" Einstein rings: Where to see yourself
A very interesting set of Einstein rings are the ``self" Einstein rings, where observers can see themselves. The most well known of these can be seen when the observer is located at the photon sphere. There observers can simply look along the photon sphere, where light travels in a circle, and see the backs of their heads! [11,25] All observers in the presence of a sufficiently compact lens, however, can see themselves. Here, light can leave the observer, travel around the lens and return to the observer to be viewed. Observers would see themselves as a series of Einstein rings. The more times light can circle the lens and return to the observer, the more "self" images the observer can see. For a lens compact enough to have a photon sphere, observers can, theoretically, see themselves in every self Einstein ring: an infinite number of times.
Amusingly, there is only a single case where observers can see only a single image of themselves - and this is the case that is well known [25] - when observers are at the photon sphere! Here all the self Einstein rings actually merge with the photon sphere to form a single observer image.
Observers who see themselves would be viewing themselves with high amplification. This is because the self images observers would see would be on or near Einstein rings - which carry the highest amplifications. Therefore gravity has become a powerful microscope! When at the photon sphere observers can microscopically view the backs of their heads, and when far away observers can microscopically view their own eyes. This is because the light that returns to the observer has left on a nearly radial trajectory - and the part of the observer most nearly radial is the observer's own eye. When close to and inside the photon sphere, observers can inspect annular rings on their heads (or spacecrafts).
IV. JOURNEY TO A HIGH GRAVITY STAR
A fairly detailed description of the distortion effects a space-traveler (or camera) would see on a visit to a high gravity star is now possible. The case that will be described first will be a trip to a "normal" neutron star: one with a currently popular equation for the interior structure of the star. This star is not dense enough to have an event horizon or photon sphere.
The second case that will be described is that of visiting a black hole. This case is more complex in that many bound and unbound photon orbits exist near the black hole. There is, however, a somewhat simpler aspect to describing this case than the previous one in that one does not have to track surface feature distortions for a black hole.
The third and last case that will be described is that of visiting a ultracompact neutron star - one with an extreme equation for its interior structure that allows a mean density so high the star has a photon sphere. This is the most complicated case of all to describe as it involves all three types of photon orbits described above as well as requiring a description of both the sky and surface feature distortions.
To more clearly delineate what the viewer would see, a set of computer generated figures were created that document the distortion effects in terms of familiar icons. In these illustrations, the sky in the background behind the high gravity star was taken to be the night sky as viewed from present-day earth. More specifically, the background sky is taken from the Bright Star Catalogue, [26] allowing all stellar images as dim as 5th magnitude to be seen, and stellar images as dim as 7th magnitude may be amplified into visibility. In the two cases of neutron stars, a map of the earth was projected onto the surfaces of the stars and allowed to distort. These figures are, in many aspects, fully general relativistically correct. The resolution of the figures is about 3 arcminutes (0.05 degrees).
Stellar image brightnesses are shown by the area the stellar image takes on the plots: the area is directly proportional to the flux the observer would receive from the image. It was impossible to change the pixel brightness, so many of the single pixel images would actually be seen dimmer than shown in the figures. Stellar images were allowed to get brighter or dimmer by angular amplification effects, but time induced amplification effects have been suppressed.
Note that for A_angular > 1 the stellar image flux would actually be seen as an increase in angular area of the image, so that the amplified angular area of the stellar images in the computer generated plots are, in this sense, realistic. However, the distortions in the amplified images would not be readily observable, as these background images would be unresolved by the viewer and hence indistinguishable from point sources. A small amplification would not cause the image to be resolved. Stellar images will therefore always be depicted as circles, even when they undergo angular amplification, as these convey best the idea of an unresolved point sources.
Only the two brightest images of all sources were tracked by the computer programs used. All stars originally 5th magnitude or brighter are plotted as secondary images, no matter their magnitude after gravitational distortion. Stars originally 5th magnitude are only plotted as primary images, however, if their final post-lensed magnitude was 5 or brighter. Higher order images undergoing larger angular amplification could potentially be visible but one would need significantly better angular resolution so see them (the only exception to this will be Fig. 2p), so they will be suppressed. An angular amplification limit of a factor of 100 was placed on all images for plotting purposes.
The hypothetical "camera" used in the simulations is somewhat fanciful but has several defining characteristics. First of all the camera is asymptotically small so that no general relativistic light bending effects are important over the length of the camera. The camera's field of view is 90 degrees across the middle of the picture. Lastly, the illustrations that follow, produced by the "camera," have been "flat-fielded" so that angular area on the spherical sky is directly proportional to spatial area on the flat page.
And always i must please you monkeys with proof of research
Frontiers of General Relativity (Little, Brown, and Company, Boston, 1977), pp. 120-150.
R. L. Forward, Dragon's Egg (Ballantine, New York, 1980).
L. Niven, Neutron Star (Ballantine, New York, 1968), p. 9.
C. Ftaclas, M. W. Kearney, and Pechenick, K. R., "Hot Spots on Neutron Stars. II - The Observer's Sky," Astrophys. J. 300, 203-208 (1986).
H.-P. Nollert, H. Ruder, H. Herold, and U. Kraus, "The Relativistic `Looks' of a Neutron Star," Astron. Astrophys. 208, 153-156 (1988).
C. T. Cunningham, "Optical Appearance of Distant Observers near and Inside a Schwarzschild Black Hole," Phys. Rev. D. 12, 323-328 (1975).
J. Schastok, M. Soffel, H. Ruder, and M. Schneider, "Stellar Sky as Seen From the Vicinity of a Black Hole," Am. J. Phys. 55, 336-341 (1987).
H. C. Ohanian, "The Black Hole as a Gravitational `Lens'," Am. J. Phys. 55, 428-432 (1987).
J.-P. Luminet, "Image of a Spherical Black Hole with Thin Accretion Disk," Astron. Astrophys. 75, 228-235 (1979).
W. M. Stuckey, "The Schwarzschild Black Hole as a Gravitational Mirror," Am. J. Phys., submitted (1992).
L. Palmer and W. Unruh, shown at a Texas Symposium on Relativistic Astrophysics in the late 1970s.
B. R. Iyer, C. V. Vishveshwara, S. V. Dhurandhar, "Ultracompact (R less than 3 M) Objects in General Relativity," Class. Quant. Grav. 2, 219-228 (1985).
K. Schwarzschild, "Ueber das Gravitationalsfeld einer Massenpunktes nach der Einsteinschen Theorie," Sitzunsgsber. dtsch. Akad. Wiss. Berlin, 189-196 (1916).
A. Einstein, "Die Grundlage der allgemeinen Relativitatstheorie," Ann. Phys. 49, 769-822 (1916).
Take it to the Jesus bloody christ thread, religion-boys. This here's science country.
To spark of a new and non-philisophical debate, heres a question for qualthwar : if nucleons ( neutrons and protons ) are made of quarks, what holds the quarks together? and could you combine different quarks with different properties to make new nucleons and hence new "charmed" or "strange" elements and substances?
To spark of a new and non-philisophical debate, heres a question for qualthwar : if nucleons ( neutrons and protons ) are made of quarks, what holds the quarks together? and could you combine different quarks with different properties to make new nucleons and hence new "charmed" or "strange" elements and substances?
lucifix
I did something m0tarded and now I have read only access! :(
lucifix
I did something m0tarded and now I have read only access! :(
I know I am a bit late for the party, so I just throw in my 2 cents (euro) randomly
(...) accident post
(...) accident post
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OK, I am late for the party, so I'll just throw in my thoughts at random
Well, assuming Gravity does affect things over a 4dim distance...
There is, in fact a 4-dimensional space definition, called space-time, in physics. There are more actually, but the 4dim space is obviously easier to work with ans useful for solving many problems... much like Newton mechanics are sufficient for solving many problems, even though it's proven "wrong" and there are now relativistic mechanics, which are more complicated.
Anyway.
Of course the center of 4dim space isn't inside our perception 3dim space. Where did BB occur? In the past, obviously. Of the universe is entirely homogeneous (which doesn't have to be true) the center lies at the point where BB occurred, and that's a long way off in space-time
But our 3dim space also has a center. Why? Well... supposedly, at BB the whole universe started to expanded more or less symmetrical. It is according to common belief NOT endless. So, what we have in 3dim space is a sphere with a limited radius. Obviously, such sphere has a center.
Well, that'S only a theory. I had a prof who's kinda expert in super-string theory and who stands for 11 dimensions. Claims certain observations in quantum-physics experiments only work for 11 dimensions. But then, those theories are a kind of sophisticated guessing really. I mean, Newton was wrong, but nor entirely so. What he found out corresponded with his limited observations. As I doubt we're now capable of seeing everything it may be similar with the best of our theories.
Why, yes. But that's just what we can perceive. We cannot know about what is outside the universe. The BB obviously marks a point of change. Before, our universe wasn't, afterwards it was. Or maybe 'beyond' this point as 'before' and 'after' are defined by our concept of time, and this point lies outside.
The question is wherein this point does lie. In another dimension maybe. But that is beyond our understanding. Assuming there is something outside our universe, then wherein is this something etc. etc.
Now that's a bit too simple, isn't it? And as I said above, time is a part of space.
Although I gotta admit I got some problems with understanding the space-time theory.
I mean, do we actually move inside space-time, or does all matter stretch from the beginning to the end like threads, intervoven according to what we perceive as movement? Is a single point of matter (hey I'm talking abstract 'k?) that is standing still like a straight thread and a moving one just a thread with curves?
Or is it so that everything exists in a point in space-time, moving in it? But if so, how is this movement defined? Movement (speed) in space is "change of place over time" but in space-time there would have to be a kind of super-time or it'd be static.
I doubt the second theory though. Antimatter supposedly moves backwards in time... but if it existed only in a single point in space-time, it would "rush" past our present and we couldn't perceive it, because a moment later, it wouldn't be there anymore, but in the past. Also, what would make everything move at the same "speed" instead of things rushing ahead or staying behind? There might even be matter at a point that has no distance in space from your position but only in time. Now imagine that matter not moving... think about the impact If you survived, you might be halted in time, the rest of the world you know rushing past you. Of course, this would make you end up in an empty space. j/k
So the point in space-time theory is out from my point of view. Everthing exists at every time. Now you know why time-travel is completly impossible: You are already there, you just cannot perceive it.
You can feel wind, see it's effects on the environment and you can smell smell. Noone said anything about seeing things. It's about perception. You can perceive time. How? Because things change. Even if you sit unmoving and try to think of nothing, there are changes. The more conscious you perceive them, the slower time seems to pass. If you are distraced, time seems to rush past.
So, we define time by changes we perceive. But as stated before, there are dimensions we cannot perceive. We cannot perceive changes in those dimensions. So we do not notice movement in those dimensions even if it occurs.
Oh, I think I'll stop here to catch my breath
Well, assuming Gravity does affect things over a 4dim distance...
There is, in fact a 4-dimensional space definition, called space-time, in physics. There are more actually, but the 4dim space is obviously easier to work with ans useful for solving many problems... much like Newton mechanics are sufficient for solving many problems, even though it's proven "wrong" and there are now relativistic mechanics, which are more complicated.
Anyway.
Of course the center of 4dim space isn't inside our perception 3dim space. Where did BB occur? In the past, obviously. Of the universe is entirely homogeneous (which doesn't have to be true) the center lies at the point where BB occurred, and that's a long way off in space-time
But our 3dim space also has a center. Why? Well... supposedly, at BB the whole universe started to expanded more or less symmetrical. It is according to common belief NOT endless. So, what we have in 3dim space is a sphere with a limited radius. Obviously, such sphere has a center.
Well, that'S only a theory. I had a prof who's kinda expert in super-string theory and who stands for 11 dimensions. Claims certain observations in quantum-physics experiments only work for 11 dimensions. But then, those theories are a kind of sophisticated guessing really. I mean, Newton was wrong, but nor entirely so. What he found out corresponded with his limited observations. As I doubt we're now capable of seeing everything it may be similar with the best of our theories.
Why, yes. But that's just what we can perceive. We cannot know about what is outside the universe. The BB obviously marks a point of change. Before, our universe wasn't, afterwards it was. Or maybe 'beyond' this point as 'before' and 'after' are defined by our concept of time, and this point lies outside.
The question is wherein this point does lie. In another dimension maybe. But that is beyond our understanding. Assuming there is something outside our universe, then wherein is this something etc. etc.
Now that's a bit too simple, isn't it? And as I said above, time is a part of space.
Although I gotta admit I got some problems with understanding the space-time theory.
I mean, do we actually move inside space-time, or does all matter stretch from the beginning to the end like threads, intervoven according to what we perceive as movement? Is a single point of matter (hey I'm talking abstract 'k?) that is standing still like a straight thread and a moving one just a thread with curves?
Or is it so that everything exists in a point in space-time, moving in it? But if so, how is this movement defined? Movement (speed) in space is "change of place over time" but in space-time there would have to be a kind of super-time or it'd be static.
I doubt the second theory though. Antimatter supposedly moves backwards in time... but if it existed only in a single point in space-time, it would "rush" past our present and we couldn't perceive it, because a moment later, it wouldn't be there anymore, but in the past. Also, what would make everything move at the same "speed" instead of things rushing ahead or staying behind? There might even be matter at a point that has no distance in space from your position but only in time. Now imagine that matter not moving... think about the impact
If you survived, you might be halted in time, the rest of the world you know rushing past you. Of course, this would make you end up in an empty space. j/kSo the point in space-time theory is out from my point of view. Everthing exists at every time. Now you know why time-travel is completly impossible: You are already there, you just cannot perceive it.
You can feel wind, see it's effects on the environment and you can smell smell. Noone said anything about seeing things. It's about perception. You can perceive time. How? Because things change. Even if you sit unmoving and try to think of nothing, there are changes. The more conscious you perceive them, the slower time seems to pass. If you are distraced, time seems to rush past.
So, we define time by changes we perceive. But as stated before, there are dimensions we cannot perceive. We cannot perceive changes in those dimensions. So we do not notice movement in those dimensions even if it occurs.
Oh, I think I'll stop here to catch my breath
the only thing i copied were the equations and those came outa a book! i scaned them, uploaded them to my web space, and linked them.Originally posted by Rukee
GoAt.......
The new "copy and paste" king!!
while i hardly understood some of what the mathmatical part meant, the rest i had fun with, seeking help form my boss and fellow employees.
MS word is great for spell checking and grammer correcting.
But I do not believe those elements consist of "new" quarks as Frosty seems to imply...
But apart from that you're right of course... although it may be possible there are stabile elements that are unlikely and therefore haven't been discovered yet. I know scientists used to believe they could some with a very high index (term?)
But I do not believe those elements consist of "new" quarks as Frosty seems to imply...
But apart from that you're right of course... although it may be possible there are stabile elements that are unlikely and therefore haven't been discovered yet. I know scientists used to believe they could some with a very high index (term?)
The new elements are just bigger elements with unstable nuclei. However, there is a theory that if you get a nucleus big enough, it will become stable in certain conditions...islands of stability in the periodic table. So there could be new elements that are stable, and would not occour naturally, but it is unlikely we could make them.
But apart from that you're right of course... although it may be possible there are stabile elements that are unlikely and therefore haven't been discovered yet. I know scientists used to believe they could some with a very high index (term?)
The new elements are just bigger elements with unstable nuclei. However, there is a theory that if you get a nucleus big enough, it will become stable in certain conditions...islands of stability in the periodic table. So there could be new elements that are stable, and would not occour naturally, but it is unlikely we could make them.
I know... but with "charmed" and "strange" elements you meant elements with new Hadrons and/or Leptons in the atoms, no?
Like a muon-atom, for instance. Oh, well. This, at least, has indeed been fabricated.
No copy and paste, huh? That sure is funny how you and your "boss and fellow employees" came up with the exact same text as this page on the Internet:
http://www.phy.mtu.edu/bht/nslens_intro.html
ROFLMFAOTLSM!!!!!!!!!!!!!!!!!!!!!!!!
damn, i guess i didnt cover my tracks good enough.
but next time, YOU WILL NOT BEABLE TO TRACE IT BACK
damn, i guess i didnt cover my tracks good enough.
but next time, YOU WILL NOT BEABLE TO TRACE IT BACK
Doesn't matter, i didn't believe you before, which is why i located what i did, and i sure won't believe anything else you try to pawn off as your own.
