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.
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.
Common objections covered, plus another way to see the violation
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:
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 = 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:
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, which is a violation of its own EP.
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.