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Why Spaghettification Does Not Occur

by Ann Hall Kitzmiller

 The fate of matter falling into a black hole has fascinated professional and amateur cosmologists for decades. Stephen Hawking theorized in his book, A Brief History of Time, that if an astronaut fell into a black hole he would be stretched "...like spaghetti..." and torn apart. Quantum Mechanics added a twist to the issue by proposing that the astronaut would be shrunk into foam. In this article, I demonstrate that the mathematical requirements for spaghettification are incompatible with the spacetime around a black hole; therefore, spaghettification does not occur. I conclude that mass approaching a black hole eventually turns into plasma and then into a stream of elementary particles.
Eq. 1

Equation 1 is the general equation used to fnd the gravitational force between two bodies in space. G is the gravitational constant, m1 and m2 are the masses of the two bodies and r2 is the square of the distance between them.
 Let's begin with today's conventional wisdom. Gravity is the force which causes objects to move toward each other. A manifestation of this effect occurs when a small mass moves toward a large one, such as when an apple falls to the Earth. Equation 1 shows the mathematical formula used to evaluate the force of gravity. G represents the gravitational constant (6.67 x 10 -11 N . m2 / kg2), m1 and m2 are the masses of the two objects, and r2 represents the square of the distance between the two objects. Gravity becomes stronger as the value of r becomes smaller. The converse is also true. Since the equation divides the product of the masses by the square of the distance between the masses, gravity becomes weaker as r becomes larger. In other words, gravity is stronger when objects are close together and weaker when the objects are farther apart.

Figure 1. The spaceman is falling into the black hole feet first. The distance from the center of the black hole to his feet is d and (d + x) is the distance from the black hole to his head. Spaghettification is the concept that the force of gravity is greater on his feet than on his head. This difference will lengthen him, eventually pulling him apart.
 Figure 1 shows a picture of a spaceman (we'll call him Rick) and a black hole. To evaluate the force of gravity on Rick we use Equation 1. G is the same gravitational constant. m1 and m2 are the masses of Rick and the black hole, respectively. But what distance do we use for the distance between Rick and the black hole? Since we are concerned with spaghettification we want to measure the effect on his feet and his head so we need to evaluate the formula twice. We use d for the distance to his feet and (d + x) for the distance to his head. Equation 2 evaluates the force on Rick's feet by using d 2 in the denominator. Equation 3 evaluates the force on Rick's head by using (d + x)2 in the denominator. The difference between these two equations, as shown in Equation 4, represents the extra force on Rick's feet which is responsible for the lengthening action assumed by spaghettification. G, m1, and m2, remain the same as Rick falls into the black hole. Only d and x change. This means that Fs decreases if d gets larger. So, the farther away Rick is from the black hole, the smaller any force that might pull at his body.
 Scientists use various measurement techniques, such as grids, to facilitate their calculations. Figure 2 places an equal distance unit grid over the masses pictured in Figure 1. This grid is essentially a Cartesian coordinate structure. On our grid, the distance between the feet and the black hole is about 19.5 units and the spaceman is about 17.5 units tall. To evaluate Fs we replace d and x in the equation with 19.5 and 17.5, respectively. Of course, if our grid was the proper scale, then in the actual calculations, d would be huge, perhaps 160,000 kilometers and x would be small, about two meters. By solving equations like this, astronomers conclude that the tidal force on the spaceman is very strong; strong enough to pull him apart.

Equation 2 computes the force on the spaceman?s feet.
Equation 3 computes the force on his head.
Equation 4 takes the difference between the other two equations to find the tidal force on his body that causes spaghettification.
 This greater effect of gravity on the parts of Rick's body which are closer to the black hole is called the tidal effect. The only difference between the tidal effect that we feel on Earth and what Rick feels from the black hole is a matter of degree. Black holes at the centers of galaxies have the gravitational effect of many millions of suns. In our example, we will assume that we are dealing with a black hole with the gravity of three solar masses. In any case, the mass of the black hole is much larger than the mass of a man, so the tidal effect in general is more extreme. Astronomers conclude that the closer Rick gets to the black hole, the stronger the pull on his feet relative to his head. And, therefore, as he falls in, his feet will fall in faster than his head and he will be elongated. Mathematically, astronomers say that x increases. This tidal effect, computed by these equations, is the spaghettification effect that astronomers use to describe what happens when masses fall into a black hole.
 The use of the gravitational force equation is valid only when the equation's unit size remains constant throughout the entire event being evaluated. The reason for this restriction is that all force equations, including the simplest f = m a (i.e. force equals mass times acceleration), and the equations we've already examined, include an acceleration computed using the derivative process from Calculus. Constant unit size is an inherent requirement for a valid derivative. The magnitude and type of the units cannot change while evaluating an equation using those units. The entire mathematical structure leading to the conclusion called spaghettification hinges on the requirement of equal length distance units on a grid structure.
 Astronomers use three dimensional Cartesian-style coordinate systems, referred to collectively in the rest of this document as the 3-D grid, to locate different celestial objects from our vantage point on Earth. They impose this 3-D grid on the sky so that they can plot the current and future locations of objects they wish to view with their telescopes. Figure 3 shows the positions of Earth, a black hole, and a spaceman on an equal distance unit 3-D grid. The actual coordinate systems astronomers use, such as right ascension and declination, and azimuth and altitude, are similar to systems of latitude and longitude and don't really form perfect squares. They do, however, still meet the critical requirement of the 3-D grid. Specifically, the size of their distance units (degree, kilometer, light year, etc.) is constant and unchanging.

Figure 2. An equal distance unit grid has been superimposed on the spaceman and black hole from Figure 1. This grid uses the classical Cartesian coordinate system that is required to evaluate the equations used in spaghettification.
 A black hole is the result of the collapse of a star, or collection of stars, into a point called a singularity. The mass doesn't change in amount during the collapse, but is crammed into a size so small that the spacetime around it is extremely dense and warped; so severely warped that photons cannot escape from this compressed zone. The existence of mass causes the contraction of space and dilation of time around the mass and is a cumulative property. The spherical edge of the zone is called the event horizon. Since photons can not reach us from inside the event horizon it appears black; hence the name black hole. The spacetime warpage immediately around the event horizon acts like a lens, bending gravity, and therefore bending light from any stars behind the black hole. This generates arc-like images of those stars around the edge of the event horizon.
 Spacetime warpage is not limited to large dense masses like black holes. Any mass, large or small, has a warping effect on the spacetime around it. A small mass like an apple may have a negligible, but never zero, effect on spacetime. In contrast, a large mass like a black hole has a major effect on the spacetime around it.
 The contraction of space changes the size of what ever distance units are being used to measure it. Imagine that you take a wet kitchen sponge and place marks at one inch increments along its length. Compress that sponge in your hand and your one inch measurements will suddenly become inaccurate, and much smaller than one inch.
 Figure 4 illustrates two different ways to organize the same space. Figure 4-A is an example of how astronomers think of space when calculating spaghettification. This represents the concept of all distances being equal that they impose on the sky.
Figure 3. A 3-D Cartesian grid (not to scale) plots the relative locations of Earth, a black hole, and a spaceman. The apparent change in unit size is due to Euclidian perspective.
 Does the spaceman falling into the black hole scenario satisfy the gravitational equation's requirement of identical length distance units? To answer this question, we will explore what a black hole does to the space around it.
 However, current thought on mass and spacetime defines Figure 4-B as the preferred representation for the space around a black hole (or any large mass). A spaceman at the red dot in Figure 4-B would exist in a spacetime in which distance units closer to the black hole would be measured as progressively smaller than the distance units farther away from the black hole. Our spaceman, Rick, would not perceive this or be able to measure the effect directly. The two-meter stick in his hand would shrink if it were pointed toward the mass since the distances between and within the atoms in its fibers would be smaller. He would think that the total length represented by the stick was unchanged. If Rick pointed the two-meter stick away from the black hole, the distance between and within the atoms would be greater with the increased distance from the black hole. And again, Rick would perceive the new elongated stick as two meters long. If Rick were moving around the two white frameworks, shown in Figure 5, he would conclude that they were edges of cubes because his frame of reference would be skewed, making the edges appear equal in length and straight instead of curved.
 Let's suppose that Rick wanted to know how many meters there were between his location and the center of the black hole. We will assume that he could attempt this feat by tethering a long string to a buoy which is in orbit around the black hole. The loose end of the string would "fall" into the black hole. Then his task would be to use his measuring stick to find the length of the string between the buoy and the center of the black hole. Because a black hole contracts the space around it, as in Figure 4-B, the number of meters between the buoy and the center of the black hole would be huge. In fact, there would be an immeasurably large number, not equal to infinity, of meters between the buoy and the center of the black hole. Rick would never be able to complete his task.
 It would be impossible to evaluate the previous gravitational force equations with the huge number of meters Rick would find. The fact that this distance would be gigantic means that the value of the force would be very small. The bigger the number reached the smaller the force would become and eventually the value of the force would approach zero, making the force equations useless in the context of warped space. This is very consistent with Einstein's idea that the force of gravity should be replaced with spacetime curvature, and that all of the motions of masses can be explained by the warping of spacetime. It is also consistent with the notion that the force equations can only be evaluated on the 3-D grid and not in warped space.
 I have demonstrated that the spacetime conditions required to evaluate the gravitational force equations do not exist around a black hole. Astronomers evaluate force equations in these inappropriate circumstances and then draw the invalid conclusion that spaghettification occurs. Spaghettification does not happen to the spaceman falling into the black hole.
 Let's explore what would actually happen to the spaceman. The black hole we will use in our scenario is the result of the collapse of a star of about three solar masses in size. The collapsing star would get smaller and smaller and blink out leaving a zone in its place which would appear to us as a black sphere.
 Imagine a spaceship orbiting the black hole and exploratory spacemen taking measurements of the black hole's effect on its local spacetime. This is not as easy as it sounds since they and their measuring devices are composed of elementary spacetime particles and are subject to the same changes that they are trying to measure. They cannot tell how the space unit or time unit is changing by direct measurement. They have to infer the changes from the behavior of other masses.
 Suppose that our spaceman, Rick, is part of that scientific mission to explore spacetime around the black hole. He and his buddies, Alan and Geoff, are part of the crew of the spaceship Intrepid. To accumulate the necessary data, Rick and Alan put on spacesuits so that they can leave the Intrepid and drift toward the black hole. Figure 6 diagrams the configuration they hope to achieve. Each man on the EVA (extra-vehicular activity) has a clock on his chest which Geoff can read from the Intrepid. Each also carries a two-meter stick to check his height and an LED readout on his wrist to report his pulse rate.
 All three men set their respective clocks to zero as Rick leaves the spaceship. Geoff promptly begins recording the elapsed time from the three clocks. Alan leaves the ship on schedule when both his clock and the Intrepid's clock reach 10 minutes.
Figure 5. Space is contracted in the direction of mass. A spaceman floating around the two white frameworks would conclude that they were cubes because his measuring stick would mimic the spacetime of the cubes.
 Rick sees Alan exit the hatch and wonders why the plans have changed. He knows that Alan was supposed to leave ten minutes after he did, but, when Rick checks his clock, it only reads 9 minutes. The dilation of time already affects him because he is closer to the black hole than are Alan and the Intrepid. Rick checks his two-meter stick and finds that his height is the same as on the Intrepid. Rick senses no change in his surroundings and concludes that the plans have changed.
 Actually, the plans have not changed. Alan did exit the spaceship 10 minutes after Rick did; 10 Intrepid minutes. Rick's space unit had contracted and his time unit had expanded since he left the Intrepid. He is actually shorter than when he left the Intrepid and his pulse rate is slower, but his measuring devices are affected in the same way so he can't identify any change. In fact, all of his bodily functions are slower; he blinks more slowly, he breathes more slowly, and his synapses fire more slowly than when he was on the Intrepid. Rick experiences his local time as "normal" because his measuring devices are subject to the changing spacetime. Rick notices that Alan's motions seem more rapid than normal. However, when Rick looks at Geoff on the Intrepid, he thinks that Geoff's behavior is even more peculiar. Geoff seems to be sped up and running around frenetically like the Keystone Cops of silent movie fame.
 Alan is in a very interesting position between the Intrepid and Rick. When he looks toward Rick and the black hole, he sees Rick moving slowly due to the dilating time unit. Rick eventually appears frozen in space because his movements are so slow. He also seems smaller than expected. When Alan looks toward the Intrepid, he sees that motions appear more sped up than normal. Alan always sees his own local spacetime as normal and others appear to behave oddly.
 From Geoff's point of view on the Intrepid, Rick appears to be moving progressively slower, so much slower that Rick eventually appears frozen in space. As Alan moves closer to the black hole Geoff sees him slowing down too, just like Rick. Geoff sees Rick's behavior as more exaggerated than Alan's because Rick is closer to the black hole and he therefore experiences a more severe warping of spacetime. Geoff notices that Alan reacts in a similar manner to Rick at the same distances from the ship. Both astronauts appear to Geoff to become increasingly smaller as the EVA proceeds.

Figure 6. Geoff watches Alan and Rick from the spaceship. Rick and Alan are smaller than they were on the ship due to the contraction of space around the black hole. Rick is smallest because he is closest to the black hole.
 Geoff sees his friends shrink in size for two reasons. First, is the familiar illusion of size reduction due to Euclidian perspective. Geoff is familiar with seeing the two crewmen appear smaller as they walk away from him. This is because the angle they subtend in Geoff's field of vision gets smaller. They do not really change size. The second reason they appear smaller is that they are actually reduced in size due to the effect of the black hole. The warpage of spacetime around the black hole causes the elementary particles, such as electrons and quarks, in Rick's and Alan's bodies to be closer together.
 After 80 minutes have elapsed on the Intrepid Geoff has recorded the times shown in Figure 7. The Intrepid remained in orbit a comfortable distance away from the black hole so there was no change in the time or space units for Geoff during the experiment.
 All of these observations ignore the fact that information transfer takes place at the finite speed of light. We assume that the information is transferred instantaneously.
 Alas, poor Rick can't get his jet pack to ignite and he continues to drift toward the black hole. From his point of view he functions normally. He thinks he is still as tall as he was on the Intrepid and he thinks that his clock is still ticking at the same rate. As he looks toward Alan and the Intrepid, he sees Alan's jet pack fire and watches Alan move toward the spaceship and enter the hatch. As Alan closes in on the ship, Rick thinks that Alan's motions speed up until, as Alan enters the hatch, Alan seems as frenetic as Geoff. Rick watches the crew of the Intrepid move about so rapidly that they become blurred images. Then the spaceship abruptly blasts out of orbit and disappears almost instantly. Rick is stunned that they abandoned him so quickly.
Clock Readings (minutes)
RickAlanIntrepid
000
91010
171920
242730
303440
354050
394560
424970
445280
Figure 7. Time passes slowest for Rick because he is closest to the black hole. (Numbers are for illustration only.)
 The two astronauts attempt to fire their jet packs to head back to the spaceship. As Alan jets toward the ship he observes that Geoff's behavior seems less frenetic and by the time he reaches the ship, Geoff is behaving in what Alan would call a "normal" manner. This is because now both Alan and Geoff have equal-size space and time units.
 However, in Intrepid minutes, the crew spent weeks trying to figure out how to help Rick. When they looked at Rick, he did not appear to move and his clock ticked only once during the weeks they spent trying to solve his problem. As Rick continued to drift closer to the black hole, the crew eventually realized that Rick was beyond their help and the Intrepid regretfully slowly left.
 Rick finally gives up trying to get the jet-pack to work and looks around to evaluate his situation. Even though the observers on the Intrepid saw Rick as frozen in space, passage of time that Rick experiences seems normal to him. His feet will continue to contract and to move closer to his head because the space in between his head and feet is contracting (see Figure 8-B). Returning to Equation 3 for a moment, Rick's height, x, is decreasing, which is directly opposite to the result spaghettification predicts (see Figure 8-A). The shrinkage in his feet will continue to be proportionally greater than the shrinkage in his head (see Figure 9). He will not be able to measure the shrinkage because it will affect his measuring stick also. Ironically, he might look down at his feet and, seeing them to be smaller than he is used to, conclude that they are farther away. This is what someone used to Euclidean perspective and expecting Spaghettification would think.
Figure 8-A. This is how Rick sees himself. It is also how the proponents of spaghettification see space.
Figure 8-B. This is how warped space really affects Rick's body. It is greatly exaggerated to demonstrate the concept.
Figure 9. As Rick falls feet first into the black hole the warpage increases on his body. Notice that as he progresses from the first frame to the fourth frame his head shrinks by 30%. However, his feet shrink by 55%. His reduction in size is due to the contracting space between his elementary particles. These ratios pertain only to this graphic and are presented just to illustrate the idea.
 Unfortunately Rick cannot fall into the black hole forever without severe consequences (see Figure 10). As his spacesuit and body shrink due to the contracted space, their electrons will react to the cramping by emitting photons. He will glow. His life support will fail when the warped space compromises the integrity of his spacesuit. His molecules and atoms will become disrupted. In a brief flash of light, the immensely warped space will convert Rick into plasma. The plasma will continue to emit photons, eventually becoming a stream of elementary particles (electrons, neutrinos, quarks, etc.).

Figure 10. Rick continues to shrink until his elementary particles, which become severely crowded, emit photons. Whatever is left of Rick is beyond the event horizon and forever out of our view. Rick's long journey to the black hole would actually take many of our centuries.
 I have demonstrated that there is a fatal flaw in the concept of spaghettification. An important assumption underlying the spaghettification scenario prevents it from being an accurate description of what happens to the spaceman. The force equations we examined (Equations 1 - 4) all require that space be divided up into distance units of equal size. In fact, equal-size units are necessary for all equations using derivatives or integrals. Only if these units are of equal length can we define the distances necessary to properly evaluate the forces described in these formulas. Since these conditions are never met around a black hole, we must conclude that spaghettification does not occur. In fact, we conclude that the actual conditions around a black hole convert any mass falling into the black hole to plasma and then into a stream of elementary particles, rather than stretching or ripping the mass apart.
 

Notes

1. Hawking, Stephen, A Brief History of Time, updated and expanded 10th anniversary ed. (New York: Bantam Books, 1998), p. 90.
2. Tipler, Paul A. and Ralph A. Llewellyn, Modern Physics, 3rd ed. (New York: W. H. Freeman & Co., 2000), p. 108.
3. Harrison, Edward, Cosmology, The Science of the Universe, 2nd ed. (Cambridge, U. K: University Press, 2001), pp. 224-233.

References and Suggestions for Further Reading

 Much of the background material I discuss has been published widely enough that it is accepted as common knowledge in the scientific community. I have included my favorite references below. In particular, Harrison's Cosmology, The Science of the Universe is a first-rate reference for Cosmology in general.
Chaisson, Eric and Steve McMillian. Astronomy Today, 4th ed. New Jersey: Prentice Hall, 2002.
Harrison, Edward. Cosmology, The Science of the Universe, 2nd ed. Cambridge, U. K: University Press, 2001.
Hawking, Stephen. A Brief History of Time, updated and expanded 10th anniversary ed. New York: Bantam Books, 1998.
Kuhn, Karl F. and Theo Koupelis. In Quest of the Universe, 3rd ed. Sudbury, MA: Jones & Bartlett Publishing, Inc., 2001
Taylor, Edwin F. and John Archibald Wheeler. Spacetime Physics, Introduction to Special Relativity, second ed. New York, W. H. Freeman & Co, 1992.

Technical Editor: Debbie Ann Newman

© 2008 Ann Hall Kitzmiller