Expanding Space Obviated (Solves the Flatness Problem)

Cosmologists today generally accept the expanding space paradigm, the notion that space itself is expanding, to explain how most of the galaxies we see are receding from one another. They believe that space must expand in a universe in which most of the galaxies recede from one another, or else the universe would have a center and an edge, violating the cosmological principle. They believe that sufficiently large objects stretch or break apart, and proffer explanations as to why smaller objects, like galaxies, don’t do likewise.1 The argument below shows that the expanding space paradigm is superfluous. It shows that space need not expand to explain how most of the galaxies we see are receding from one another. It shows that our universe needn’t have a center or edge when space doesn’t expand. The argument introduces a feature predicted by general relativity, relative expansion, which is ignored by our cosmology and explains observations that purport to confirm the expanding space paradigm, including the observation that galaxies are receding from us faster than c, the speed of light.

Let an infinitely large universe whose space neither expands nor contracts be sprinkled with an infinity of galaxies. Let S be a section of this universe, and let S have the property that all of the galaxies within it approach one another. Nothing prevents this in principle. The galaxies in S can approach one another even though the space of S doesn’t contract. The size of S is arbitrary; it can have any size. Make S infinitely large, unbounded. Now all of the galaxies in this universe are approaching one another, yet this universe has no center or edge, nor is its space contracting. “Run the film backward” to visualize all of the galaxies receding from one another, with no existence of a cosmic center or edge, nor does the space of this universe expand.

The preceding thought experiment shows that an infinitely large universe can become sparser or denser without its space expanding or contracting, and without violating the cosmological principle. Then Occam’s razor strongly suggests that space doesn’t expand or contract in our own universe, which we have no reason to believe isn’t infinitely large.

When space itself doesn’t expand or contract:

  • Locally inertial frames can in principle be arbitrarily large. The Relativistic Rocket site, whose equations are those of special relativity (SR) and apply to locally inertial frames, warns “For distances bigger than about a thousand million light years, the formulas given here are inadequate because the universe is expanding.” This limitation goes away when space doesn’t expand or contract. Then:
    • Spacetime is globally (universe-wide) asymptotically flat at great distances from the galaxies and other centers of gravitational attraction. This solves the flatness problem, solving it far simpler than the inflation theory does.2, 3, 4 (Instead of adding a giant assumption to physics, like inflation theory does, we’ve removed an assumption.) And:
    • From any vantage point at an infinite distance from every center of gravitational attraction (which is to say, nowhere), every galaxy recedes slower than c, and galaxies at a limit of infinite distance recede at speeds up to c in the limit, and therefore can be infinitely redshifted in the limit and not aging in the limit.
  • Recession speeds of galaxies can in principle exceed c (without violating SR) as measured from a vantage point at a finite distance from a center of gravitational attraction (which is to say, anywhere) due to relative (depends on the observer) spatial expansion predicted by both special and general relativity, as explained at Dark Energy Obviated. This relative expansion is measurable, it adds (without limit in principle) to recession that is due to simple movement away, and it explains observations that purport to confirm the notion that space itself absolutely expands (i.e. in a way that can break sufficiently large objects).
  • No object need be out of causal contact with another object anywhere else in the universe.
  • The entire universe is observable from any vantage point, albeit objects may be redshifted out of practical observable range (e.g. perhaps a photon from a fast-receding object reaches the observer only once a year on average on the observer’s clock), providing a solution to Olbers’ paradox.
  • The terms “expanding universe” and “cosmic expansion” are still useful to refer to the recession of most galaxies from one another.
  • No explanations are required as to why galaxies and smaller objects don’t stretch or break apart in concert with cosmic expansion.

References and Notes

1. See Cosmic expansion is not to blame for expanding waistlines at http://www.newscientist.com/article/dn8082.

2. See The Flatness Problem at http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/FlatnessProblem.html.

3. See WMAP Cosmology 101: Inflationary Universe at http://map.gsfc.nasa.gov/m_uni/uni_101inflation.html. Note that the inflation theory creates a major additional problem for cosmology, namely as to the nature of the proposed inflaton field that inflated the universe. It’s easily the biggest assumption in all of the sciences.

4. Imagine a freely falling block of Swiss cheese, billions of light years wide / tall / deep, representing a section of the universe. The holes of the cheese are regions of significantly curved spacetime. At the centers of the holes are galaxies. Some of the holes may be merged into one another to form larger holes surrounding clusters and superclusters. The holes (the galaxies) are of course free to be moving relative to one another and relative to the block. The block of cheese except the holes is a free-floating region of negligibly curved spacetime, a locally inertial frame. The block can be arbitrarily large in principle; nothing prevents that possibility now that expanding space is obviated. When the block can be arbitrarily large it follows that the default curvature of the universe is zero, flat. Then no flatness problem arises when observations show that the universe might be flat.

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Dark Energy Obviated (Also Solves the Horizon Problem)

Observations show that the expansion of the universe is accelerating.1 What causes this acceleration is an unsolved problem of physics; dark energy is only an ad hoc solution that was fitted to the observational data. The argument below shows that general relativity (GR) already provides the answer, making dark energy superfluous. Note that the argument assumes that space itself is not expanding, as explained at Expanding Space Obviated.

See the equations of special relativity (SR) for a relativistic rocket. Let a rod with a proper length of ten light years float in intergalactic space. Let a rocket decelerate relative to the rod and alongside it, first passing a beacon fixed at one end of it and finally coming to rest relative to it at its other end. Let the trip take just one year as the crew measures, as the equations allow. Then the crew observes the beacon recede to a distance of ten light years in one year, at which point the rocket is at rest relative to both the beacon and the rod.2, 3 The equations can be used to show that the crew measures the beacon pass the rocket at a speed of 99.98% of c, the speed of light.4

The only way the crew can observe the beacon recede to a distance of more than one light year in one year, when its initial speed as they measure is less than c, is if it accelerates away from them as they measure (otherwise it would recede to a distance of less than one light year in one year). The rod is initially foreshortened as the crew measures due to special relativistic length contraction. As the rocket decelerates relative to the part of the rod passing by, the rod becomes less foreshortened as the crew measures; it expands in length. At a relativistic speed v (a speed close to c) relative to the part of the rod passing by, the length expansion of the rod outbalances the deceleration to make the beacon accelerate away from the crew as they measure. The difference between ten light years of recession and less than one light year of recession (the latter corresponding to their one-year trip at a speed less than c) is due to a relative (depends on the observer) expansion of the rod, and of the space containing the rod.

Remove the rod and let the rocket’s degree of deceleration vary among multiple tests. (But keep in mind that the rocket decelerates relative to the part of the now-removed rod passing by, not necessarily the beacon, which may accelerate away, even if I say “decelerates relative to the beacon” for lack of a better tangible thing to refer to.) The relativistic rocket equations can be used to show that, regardless of the degree of the rocket’s deceleration, when the speed v is relativistic the beacon accelerates away from the crew as they measure. To see this, for any acceleration a, plot the distance d / γ (the distance the crew measures) as a function of the time T (the elapsed time the crew measures) until the speed v is relativistic. Rotate the chart a half turn, so it depicts the opposite scenario, that the rocket decelerates to rest relative to the beacon, initially passing it at a relativistic speed.

image001.gif
image002.gif
image003.gif
As measured by the crew of a relativistic rocket that passes a beacon while decelerating relative to it, their distance from the beacon as a function of time, from when they pass the beacon to when they come to rest relative to it.
Let a relativistic rocket decelerate past a floating beacon and come to rest with respect to it. As the crew measures, a curve of their distance from the beacon over time, when their initial speed relative to the beacon is low (top), has a tail when it is relativistic (middle and—closer to c—bottom). Along the tail the beacon accelerates away from the crew as they measure. The degree of the rocket’s deceleration is immaterial; these charts are producible regardless of the deceleration given.

According to the equivalence principle of GR, during their trip the crew experiences the equivalent of a uniform gravitational field, like that experienced locally by someone standing on the Earth’s surface (an “Earth observer”). From the crew’s perspective the beacon moves as if it was thrown upward in a uniform gravitational field. Then the equations of motion for a locally uniform gravitational field are the relativistic rocket equations, replacing the relevant equations of Newton’s. (The top chart above plots an everyday activity on Earth, such as tossing up a ball.) Let an Earth observer launch a particle upward at a relativistic speed. The particle initially accelerates away from the observer as that observer measures, the middle and bottom charts show. Having employed only the equivalence principle and published equations of SR, locally, we now know the solution to the “keys accelerate toward the ceiling” mystery described at National Geographic News, for keys thrown upward at relativistic speed:

“When I throw my keys up in the air, the gravity of the Earth makes them slow down and return to me,” said Mario Livio, a theoretical physicist at the Space Telescope Science Institute (STScI) in Baltimore, Maryland.

But the study, along with an independent work released later the same year, showed that the expansion rate is actually speeding up.

This observation, Livio said, is as if “the keys suddenly went straight up toward the ceiling.”

(The keys in this analogy are placeholders for the high-redshift supernovae that compose the observation leading to the idea of dark energy. The only keys we’ve observed to accelerate toward the ceiling are those moving away from us at relativistic speed, if only when we drop the idea of expanding space.)

Consider an idealized case where the Earth is the sole gravitationally-significant body in the universe. Let the crew of a rocket that is hovering above the Earth at any distance from it (and therefore feeling acceleration from a thrusting engine, however slight) locally measure a particle P that recedes directly away from the Earth with high redshift, starting from when it passes right by the rocket at a relativistic speed. Again the relativistic rocket equations predict that P initially accelerates away from the crew as they measure. (Read carefully; these are local measurements made by the rocket’s crew, confined to a sufficiently small space and duration, that therefore can be validly predicted by those equations of SR.)

The crew can communicate those measurements to an Earth observer, who converts it to his or her own measurements (what he or she would measure about P while looking through a telescope) using the relative differences in the rate of local clocks and length of local standard rods that are predicted by GR between the observers. This conversion is tantamount to merely recalibrating each axis of a chart like those above, which leaves the shape of the curve unchanged and therefore results in an unchanged conclusion regarding acceleration or deceleration. For example, no matter how each axis of the middle chart above is recalibrated (to, say, change the time scale from 0 to N seconds on the rocket’s clock, to 0 to Nx seconds on the Earth observer’s clock, where x is the rate at which the Earth’s observer’s clock runs, expressed as a fraction of the rate at which the rocket’s clock runs), “acceleration away” is still indicated. Therefore the Earth observer would also measure that P accelerates away, notwithstanding the curved spacetime between the observer and P. In other words, acceleration away is predicted by GR (across curved spacetime) as well as by SR (across flat spacetime).5

High-redshift supernovae accelerating away from us compose the observation of accelerating cosmic expansion, and are proxies for P. Then there is no need for dark energy to explain that observation. Furthermore, this thought experiment solves the horizon problem by showing that exponentially rapid cosmic expansion is possible in principle, without need for the inflaton field the inflation theory proposes.6, 7 Gravity alone does the trick. As an extreme example, the proposed new metric for Schwarzschild geometry (or the Schwarzschild metric) could be used to show that in principle an observer could measure a free object recede at a rate of ten light years per year, increasing within a second to twenty light years per year or any higher rate.8 Meanwhile, causal contact between the object and the observer would be retained, just as it is between the beacon and the rocket’s crew, or between P and the Earth observer.

Recall that the argument above assumes that space itself isn’t expanding. When the expanding space paradigm is discarded in concordance with Occam’s razor, it can now be seen that three major problems of physics vanish, with no new assumptions or theories needed (including the inflation theory): the flatness and horizon problems, and the nature of dark energy. Note that observations purporting to confirm the expanding space paradigm are explained by the argument above. Space can measurably expand in a relative way (it depends on the observer). Free objects can recede from an observer at speeds greater than c without violating SR or GR, in space that isn’t expanding in an absolute way (a way that stretches or breaks sufficiently large objects). Even if cosmologists keep the superfluous notion of space itself expanding (absolute expansion), they need to add the notion of relative expansion, assuming they agree with GR.

It isn’t necessary to feel acceleration, as the Earth observer does, to observe relative expansion or acceleration away. It’s enough to be at a fixed distance from a center of gravitational attraction; e.g. freely orbiting. That’s because gravity, not non-inertial acceleration per se, causes the deceleration on which both effects depend. The space between any observer and any receding decelerating free object expands (in a relative way) as the observer measures. If the object recedes sufficiently fast (i.e. if its redshift is sufficiently high) then the relative expansion of space causes the object to accelerate away as the observer measures. It’s not just the Earth’s gravity that makes the high-redshift supernovae accelerate away as an Earth observer measures. Also contributing (at least) is the gravity imparted by the whole Milky Way.

References and Notes

1. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, A. G. Riess et al., Astronomical Journal, vol. 116, pg. 1009 (September 1998), http://arxiv.org/abs/astro-ph/9805201.

2. Herein, as in most relativity texts, we ignore the travel time of light that prevents a remote event from being seen until after it has occurred. In principle a network of observers can be set up in locally inertial frames that momentarily co-move with the rocket, so that events can be detected by equipment right next to them, and the records subsequently compiled and analyzed.

3. Though the crew observes the beacon recede to a distance of ten light years in one year, a recession rate greater than c, this doesn’t mean that Einstein’s speed of light limit is violated, for that limit applies only to the speed of objects moving in spacetime as measured with respect to locally inertial frames, whereas the crew’s frame is non-inertial. In any locally inertial frame that momentarily co-moves with the rocket, the rod and the beacon’s speed is less than c. This “greater than c recession rate” also isn’t new information: The Relativistic Rocket site reports that a rocket accelerating / decelerating at 1 Earth gravity can travel from Earth to the Andromeda galaxy, 2 million light years away and arriving at low speed, in 28 years on the crew’s clock. Then the rocket’s crew would observe a beacon floating at the midpoint between the galaxies recede 1 million light years in the 14 years after they pass it.

4. Ideally this speed is measured in a locally inertial frame that momentarily co-moves with the rocket, since the rocket is decelerating.

5. A simpler way to see this: Flat spacetime is curved spacetime with zero curvature. Then, when acceleration away is predicted for flat spacetime it is concomitantly predicted for curved spacetime.

6. See The Horizon Problem at http://archive.ncsa.illinois.edu/Cyberia/Cosmos/HorizonProblem.html.

7. See WMAP Cosmology 101: Inflationary Universe at http://map.gsfc.nasa.gov/m_uni/uni_101inflation.html.

8. The relativistic rocket equations can predict that scenario, for an observer on the surface of a sufficiently large / massive object. (Note the hockey-stick shape of the bottommost chart above.) Then it’s predictable as well by the proposed new metric for Schwarzschild geometry. The relativistic rocket equations are special relativistic equations of motion for a locally uniform gravitational field (as deduced above), and the metric approximates SR locally, and the size of “local” is unlimited in principle. This isn’t to say that the metric couldn’t predict that scenario for a non-uniform gravitational field too.

No Black Holes (Solves the Information Loss Paradox)

This argument shows that general relativity’s (GR’s) prediction of black holes violates the equivalence principle (EP) the theory postulates.

Popularly reported discoveries of black holes, rather than being confirmations of GR, are merely implied when the theory’s validity is assumed, which is to say the existence of black holes in nature isn’t proven. No direct evidence of a black hole has been found, nor definitive indirect evidence.1 Every final determination of “it’s a black hole” has been made by inputting values into an equation of GR to see whether the theory predicts that the object in question is a black hole. There’s no other way to make such a determination: by definition a black hole emits nothing, and Hawking radiation of a suspected black hole hasn’t been observed. It’s illogical to conclude that a suspected black hole must be a black hole when matter is disappearing into it, because GR predicts such disappearance as well for an “almost black hole”, a body almost but not quite compact enough to be a black hole. Moreover, GR’s equation that predicts black holes can be tweaked to not predict black holes at all, yet still agree with all relevant observations, including matter disappearing into a black sphere.

There are many valid formulations of the EP postulated by GR. The theory need only violate one of those formulations to have a major problem. Here is the EP as described by Thorne:2, 3

In any small, freely falling reference frame anywhere in our real, gravity-endowed Universe, the laws of physics must be the same as they are in an inertial frame in an idealized, gravity-free universe. Einstein called this the principle of equivalence, because it asserts that small, freely falling frames in the presence of gravity are equivalent to inertial frames in the absence of gravity.

Set up a thought experiment as follows:

  • Let X be a freely falling reference frame that is falling through the absolute horizon of a spherically symmetric, uncharged, non-rotating black hole, having started falling from rest at a great distance.
  • Let X be small enough in spacetime (that is, both spatially and in duration) that the tidal force in it is negligible.
  • Let X contain a free test particle that is above the horizon and traversing X while escaping to infinity. (GR predicts that the escape velocity above the horizon is less than c, the speed of light.)

Frame X is defined to be a locally inertial frame. According to the EP, no test of the laws of physics can distinguish X from an inertial frame in an idealized, gravity-free universe. The following laws of physics apply in an inertial frame in such an idealized universe:

  • Law J: In principle a signal is transmittable between any two spatial points, in either direction.
  • Law K: No part of the frame is off-limits to any other inertial frame.

Law J is self-evident. Law K is given by the fact that inertial frames can always be unbounded in the absence of gravity. In an idealized, gravity-free universe, inertial frames can encompass the entire universe, so they can intersect one another at every point. The EP and law K imply that any locally inertial frame that is wholly within any other locally inertial frame must be extendable to fill all of the outer frame. There can’t be any part of the outer frame that is off-limits to the inner frame.

GR predicts that nothing, not even light, can pass outward through the horizon. Then a signal can’t even in principle be sent from a point below the horizon to the escaping particle, which stays above the horizon. Then law J shows that a locally inertial frame relative to which the particle is at rest can’t even in principle extend below the horizon, including a frame wholly within frame X. Then law K is false in X, and a test of the laws of physics can distinguish X from an inertial frame in an idealized, gravity-free universe. Then GR contradicts its own EP; the theory is self-inconsistent.

Common objections covered, plus another way to see the violation

Many readers point to the negligible tidal force in X, an unavoidable difference between any locally inertial frame and an inertial frame in an idealized, gravity-free universe, as a culprit that invalidates the thought experiment above. But the tidal force can’t be the reason that law K is false in X, if only because law K is true in other locally inertial frames. Many experimental tests of the EP exist despite the tidal force in the labs in which they were performed. Those tests aren’t necessarily invalid due to the tidal force, even if their results contradict the EP. The same applies to a thought experiment that tests GR’s adherence to the EP. The tidal force is covered in greater detail below.

The thought experiment above takes place completely within the confines of frame X, which is defined to be small enough in spacetime that the tidal force in it is negligible, and the frame can be arbitrarily smaller, making it the best possible lab for testing the EP. That the particle is escaping to infinity is a process occurring within X during its limited existence. The particle needn’t reach any great distance from the black hole before the experiment concludes. Rather, during the experiment the particle may need to move only an arbitrarily small distance or for an arbitrarily short time, either as measured in X. Nothing in principle prevents a local experiment on a particle escaping to infinity.

“Tidal force” and “spacetime curvature” refer to the same thing.4, 5 A frame in which the tidal force is negligible is a frame in which spacetime is negligibly curved.

It isn’t necessary to be able to detect (by measurement) that a horizon exists within X. It’s a given in the thought experiment that a horizon is there. The laws of physics in X are affected by the presence and properties of the horizon regardless whether an observer in that frame detects the horizon.

That law K is false in frame X means that X isn’t equivalent to an inertial frame. That’s shown by unique behavior there. For example, consider the following diagram:


General relativity (GR) violates its own equivalence principle.
The frame X (defined to be a locally inertial frame) initially contains a cloud of free test particles that straddles the horizon. Let one of the particles above the horizon be escaping to infinity. GR demands that the particles below the horizon be moving inexorably toward the black hole’s singularity, whereas the escaping particle ever moves away from the black hole. Therefore it’s impossible even in principle for all of the cloud’s particles to be moving in formation, all moving away from the black hole. If X were equivalent to an inertial frame, it would be possible in principle for all of the particles to be moving in formation regardless of the velocity given for any one of the particles. This difference in the laws of physics between X and an inertial frame is a violation of the equivalence principle, which implies there is no such difference. The negligible tidal force in X can’t be the culprit, if only because such a tidal force exists in all locally inertial frames, including those for which the violation doesn’t apply.

A common misconception claims that the tidal force becomes so strong at the horizon of a black hole that no locally inertial frame can exist there. But the tidal force in X can in fact be arbitrarily weak in principle, no matter how large X is, when the mass of the black hole is variable. The general relativistic equation for the tidal force in X is (in geometric units):6

dg=\displaystyle\frac{2M}{r^3}dr

Where:

  • dg = Tidal force in a freely falling frame that falls toward and into a black hole starting from rest at a great distance.
  • M = Mass of the (spherically symmetric, uncharged, non-rotating) black hole.
  • r = Reduced circumference at the frame’s location, defined so that the circumference of a sphere at radius r is 2πr.
  • dr = The radial displacement of the frame, where r is the reduced circumference.

The reduced circumference r of the horizon is 2M. Then the tidal force at the horizon is expressed by the equation:

dg=\displaystyle\frac{dr}{(2M)^2}

Then for any given radial displacement dr of the frame, the tidal force in the frame is inversely proportional to the mass of the black hole. Then the tidal force in frame X can in principle be arbitrarily weak for any given radial displacement of it. That means the horizon isn’t a special place in regards to the tidal force.7 (It also explains how you could have crossed the horizon of a black hole while reading this sentence, without noticing anything out of the ordinary, and to reach the singularity only millenia from now.)

It cannot be that the EP is testable only impossibly within a single point in spacetime, or else it’s outside the realm of science. All ideas in science are required to be falsifiable. The EP is strictly true for only a point in spacetime, yet is testable (and has been tested to great precision) in a larger frame.8 For example, if Boyle’s law is found to not hold in a particular freely falling lab, the EP isn’t saved by the fact that the lab is larger than a point in spacetime. The EP couldn’t be saved if the tidal force in the lab didn’t alter the outcome of the experiment. The tidal force is the sole reason the principle is strictly true for only a point.9 How do we know that the tidal force doesn’t alter the outcome of the thought experiment above? We know, because law K would be true in X if not for the horizon there, and the horizon isn’t a special place in regards to the tidal force. The horizon per se has nothing to do with the tidal force.

The cloud of particles in the diagram above can’t be significantly stretched by the tidal force during the thought experiment, because frame X is defined to be small enough in spacetime that the tidal force in it negligible, which according to GR’s equation above is always possible for any given mass of the black hole, and possible as well when the frame is arbitrarily large, by letting the black hole be sufficiently massive.10 The tidal force only stretches and squeezes objects.11 It’s incapable of forcing an object to move inexorably toward a particlular destination, so it’s incapable of forcing some of the cloud’s particles to move inexorably toward the black hole’s singularity. This again shows that the negligible tidal force in X doesn’t explain away GR’s violation of the EP.

Length contraction of the cloud isn’t an issue. As measured in frame X the cloud may be length-contracted as predicted by special relativity (SR), but SR predicts that length-contracted objects always have a positive length, so the cloud can still straddle the horizon initially. The thought experiment can’t be validly refuted by employing logic that contradicts the EP or SR. The EP implies that the laws of physics in X are the same as they are in an inertial frame; therefore it can be assumed that SR applies in X, unless and until the experiment shows otherwise. An experiment purporting a violation of the EP can’t be validly refuted by employing a violation of the EP. For example, it can’t be validly said that the escaping particle must have a velocity greater than c as measured in frame X, to fail the thought experiment, because GR allows the particle to be escaping and SR says the particle’s velocity as measured in frame X is less than c. The horizon moves outward precisely at c as measured in X, according to GR, and the escaping particle outruns it in X, increasing its distance from the horizon as measured in X, but that information cannot be used to refute the thought experiment above. The information may violate the EP, but if so it shows a problem with GR rather than a problem with the thought experiment.

An objection posits that the cloud’s particles that are below the horizon could keep pace with the escaping particle only by infinitely accelerating, which isn’t possible and so the thought experiment fails. This objection blames the experiment for GR’s self-inconsistency. The particles are defined to be free test particles; therefore it’s indisputable that they are freely falling. The particles below r = 2M could in principle keep pace with the escaping particle while freely falling, as they could in an inertial frame, if only GR didn’t predict that a horizon exists at r = 2M. It’s not the experiment’s fault that GR doesn’t allow the laws of physics in frame X to be the same as those in a true inertial frame, in violation of the theory’s own EP.

Another objection goes like this: Because a Rindler horizon exists below a relativistic rocket, a horizon must also exist beneath an observer hovering above a black hole, and since SR predicts it, it must be valid.12 (Note that this thinking doesn’t attempt to directly refute the thought experiments above; it leaves them intact as sound arguments.) A Rindler horizon and a horizon of a black hole are inequivalent types of horizons. A Rindler horizon exists only in the rocket’s frame, it’s relative, whereas the horizon of a black hole is absolute, existing for all observers. The Rindler horizon doesn’t exist in the frame in which the rocket blasted off. In that frame light can pass outward through the location where the rocket’s crew reckons the Rindler horizon is, it just can’t subsequently reach the rocket to be detected by the crew.13 An exception is when the rocket’s acceleration is infinite, so that the Rindler horizon is at zero distance from the rocket. While such Rindler horizon is equivalent to a horizon of a black hole (a rocket would need infinite acceleration to hover precisely at the horizon of a black hole), it’s impossible. So SR doesn’t predict a black hole type of horizon, one through which nothing can pass outward for any observer. The reduced circumference r, at which infinite thrust is required to hover there, needn’t be at some r > 0 in a theory of gravity. It can be at r = 0 in the limit instead, so the EP isn’t violated.

Lastly, some claim, often using jargon specific to general relativity, that it’s not nearly so simple to compare frame X to an inertial frame as I’ve done herein. That contradicts the EP, so it’s unsurprising that Taylor and Wheeler explicitly disagree with it:14

No one can stop us from observing a black hole from an unpowered spaceship that drifts freely toward the black hole from a great distance, then plunges more and more rapidly toward the center. Over a short time the spaceship constitutes a “capsule of flat spacetime” hurtling through curved spacetime. It is a free-float frame like any other. Special relativity makes extensive use of such frames, and special relativity continues to describe Nature correctly for an astronaut in a local free-float frame, even as she falls through curved spacetime, through the horizon, and into a black hole. Keys, coins, and coffee cups continue to move in straight lines with constant speed in such a local free-float frame. … What could be simpler?

Frame X is defined equivalently to the frame represented by Taylor and Wheeler’s unpowered spaceship as it falls through the horizon. Then Taylor and Wheeler make it clear that no differences should be found when we compare the laws of physics in frame X to those in an inertial frame. Yet when we tested the law K defined above, a law of physics that’s true in an inertial frame, we found that it’s false in X. As the unpowered spaceship falls through the horizon it is demonstrably not a “free-float frame like any other”. GR doesn’t allow it to be a free-float frame like any other.

Conclusion

The argument above shows that GR’s prediction of black holes violates the EP the theory postulates. Being the core of the theory, the EP must survive at the expense of predictions of black holes. Since reports of black holes in nature are actually just implications of GR (no black hole has been definitively discovered), it’s reasonable to assume that they’re nothing more than a mistake of the theory.15 GR’s prediction of them leads also to incompatibility with quantum mechanics, such as described by the information loss paradox.16 See Toward a New Theory of Gravity for a proposed solution that doesn’t predict black holes, yet is confirmed by all relevant experiments as far as I know.

References and Notes

1. See the Chandra X-ray Observatory FAQ at http://chandra.harvard.edu/resources/faq/black_hole/bhole-10.html. The “unless Einstein’s theory of gravity breaks down” means that the “evidence for a black hole” depends on the validity of the theory. The observation of a black hole is not made independently of the theory.

2. Thorne, K. S., Black Holes and Time Warps, pg. 98. This quote can also be seen at http://tinyurl.com/66ygkla (note the last paragraph on that page, the last sentence in particular about a frame falling through the horizon of a black hole).

3. See Thorne’s definition of “laws of physics” at http://tinyurl.com/5raz42e.

4. Thorne, K. S., Black Holes and Time Warps, pg. 111. Read it online at http://tinyurl.com/9o2t8gt.

5. Taylor, E. F. and Wheeler, J. A., Exploring Black Holes, pg. 2-6. This chapter is online at http://www.eftaylor.com/pub/chapter2.pdf.

6. Taylor, E. F. and Wheeler, J. A., Exploring Black Holes, pg. B-20.

7. This agrees with numerous examples from reputable sources, like the following at http://casa.colorado.edu/~ajsh/singularity.html: “In a supermassive black hole the tidal forces are weaker, and you could survive well inside the horizon of the black hole before being torn apart.”

8. See tests of the EP done by the Eöt-Wash Group of the University of Washington (in a frame larger than a point!) at http://www.npl.washington.edu/eotwash/results.

9. Search http://astro.physics.sc.edu/selfpacedunits/Unit57.html for “We see that gravity is different than other forces”.

10. The above equation shows that even if frame X is light years across and lasting for years, as measured in X, in principle the tidal force in it could still be arbitrarily weak—as weak as we want to make it, by letting the black hole be sufficiently massive. As the black hole’s mass is increased the black hole gets larger, since the reduced circumference r at the horizon is always 2M. The spacetime in a given-sized X becomes flatter as the black hole gets larger, like how a given-sized patch of a sphere’s surface gets flatter as the sphere is enlarged.

11. See Gravitational Tides at http://burro.cwru.edu/Academics/Astr221/Gravity/tides.html.

12. For an explanation of a Rindler horizon, search The Relativistic Rocket at http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for “Below the rocket, something strange is happening…”.

13. This difference between a Rindler horizon and a black hole’s horizon is easy to visualize using the example at The Relativistic Rocket site.12 While a Rindler horizon exists below the star (and stays below it) as the rocket’s crew reckons, light isn’t prevented from traversing the non-accelerating frame (in which the rocket blasted off) in any direction. Light that passes outward through the Rindler horizon just can’t reach the rocket as long as it keeps accelerating. As The Relativistic Rocket site says, “But of course, nothing strange is noticed by the non-accelerating Earth observers. There is no horizon anywhere for them.” Whereas the (absolute) horizon of a black hole exists for all observers.

14. Taylor, E. F. and Wheeler, J. A., Exploring Black Holes, pg. 2-4. This chapter is online at http://www.eftaylor.com/pub/chapter2.pdf.

15. In which case objections employing Hawking radiation are moot points. If black holes are a mistake of GR, then so are evaporating black holes. Hawking radiation cannot prevent the EP from being tested in full (lest it not be science, wherein all ideas are required to be falsifiable). A logical test of any of the EP’s claims takes precedence over any ramification of Hawking radiation.

16. See Black Hole Information Paradox at http://en.wikipedia.org/wiki/Black_hole_information_paradox.