Breaking spacetime

Abstract

This note proposes a new interpretation of the hypothesized maximum force in General Relativity by drawing an analogy between spacetime and brittle materials, using Griffiths’ fracture theory at face value. It suggests that the limit for spacetime breakage at the Planck scale naturally leads to the maximum force concept. The analysis also introduces a hypothetical ‘bonding energy’ between spacetime’s fundamental building blocks. Finally, it speculates on the formation of a kugelblitz, concluding that vacuum polarization (Schwinger effect) would prevent its formation, even at Planck scales. The work remains speculative and pedagogical but may inspire further ideas.

(The article’s fifteen displayed equations are rendered as images by the publisher and do not survive text extraction; they are given below as named results in words, in the article’s own order and numbering. A few inline symbols are likewise absent and are named rather than reproduced. The exact forms are in the published article.)

1. Introduction

The idea of a maximum luminosity radiated by a highly dynamic astrophysical system, for example a collapsing or exploding star, has been contemplated long ago. This has led to speculation on the hypothesis of a maximum force allowed by General Relativity. Although this proposal has been considered by many authors, its full justification still remains controversial. Different explanations have been called upon through string theory, cosmological arguments and black holes, among others. The question remains of interest due to its relationship with modified gravity theories as well as its possible connection to the cosmic censorship hypothesis.

The purpose of this brief note is adding a different view to the existing debate. It is meant to be a pedagogical and speculative exercise without full mathematical rigour. We will make use, face value, of the Griffith fracture criteria normally applied in the context of fracture mechanics for isotropic and brittle materials. In this case, it will be applied to the ‘spacetime fabric’, as if it were a material valid within the range of applicability of Griffith’s theory. We will argue how the maximum force emerges, which then introduces a limit beyond which the ‘fabric of spacetime’ could be broken.

The authors are, of course, well aware that considering spacetime as a ‘fabric’ is no more than an analogy. Nevertheless, two remarks might be worth mentioning. The first is that the theory of elasticity in the context of general relativity was developed in the mid-twentieth century. The need for it came in the late 1950s with Weber’s bar antenna for gravitational waves, which has then continued attracting the interest of researchers. The second one is that the elasticity of spacetime has been discussed in relation to gravitational waves by today’s leading experts. For example, a question that was asked at the end of the 2018 Hamilton Lecture at Princeton, Exploring the Universe with Gravitational Waves by Kip Thorne, concerned the magnitude of the Young’s modulus of spacetime. The answer was given by Rainer Weiss to be frequency dependent and approximately 10²⁰ times the one of steel for a frequency of 100 Hz.

2. Griffiths’ fracture criteria and maximum luminosity

Let us first briefly remind the so-called Griffith’s fracture criteria. According to it, and considering an isotropic brittle material, failure occurs when a critical stress is reached and cracks propagate fully. Such critical stress is given by Equation 1, in which the material’s Young’s modulus, measured in newtons per square metre, the energy required to break atomic bonds per unit surface area created by the crack, measured in joules per square metre, and the length of the crack all enter. At a stress higher than the critical one the crack becomes unstable and catastrophically grows.

Let us now remind, also briefly, that the maximum luminosity, that is the maximum output power, radiated by a violent astrophysical object occurs when it is near its gravitational radius. It is constant regarding the nature of the system and given approximately by the well-known relationship of Equation 2, in which the speed of light and Newton’s constant appear and the luminosity is measured in joules per second.

3. An analogy for spacetime fracture

As previously indicated, we will suppose that spacetime is an ‘isotropic and brittle fabric’ and we will reason for applying the Griffith’s fracture criteria.

In the first place, we will assume a certain total elastic energy released in a volume of spacetime equal to the Planck one. The total elastic energy is given by Equation 3, in which the stress is measured in newtons per square metre, so that the elastic energy is given in joules. Let us now consider that stress is reaching its critical value and substitute Equation 1 into Equation 3 to obtain Equation 4. We also have considered the crack length equal to the Planck length in Equation 1 since we are assuming a Planck length size region.

Let us now consider that the maximum luminosity is given by Equation 2. Multiplying it by the Planck time will give us the amount of energy released in such a time in joules, which is Equation 5.

Our analogy here with spacetime fracture is that we consider the maximum possible energy released, for example by a collapsing star, in a Planck area within a Planck time duration is higher than the area’s elastic energy — the ‘spacetime elastic energy’. Thus, it could be formulated, by taking Equations 4 and 5, as the maximum released energy exceeding the elastic energy, which is Equation 6, and which also gives Equation 7 by making use of the fact that the speed of light is the Planck length divided by the Planck time.

At this point three observations are worth mentioning. The first is that the right-hand side of Equation 7 contains the expression of the postulated maximum force allowed by General Relativity, which in this brief note has been obtained without the need of any contexts mentioned in previous literature.

The second is that the hypothetical ‘bond energy’ of spacetime is calculated to be of order 10⁷⁸ joules per square metre, assuming Equation 7. Below this value, a hypothetical spacetime break could occur due to the release of elastic energy from a speculated extremely dynamic astrophysical system.

The third observation considers the proposed frequency which is dependent on the Young’s modulus of spacetime, given as Equation 8. Expression 8 allows defining of ‘axial spacetime stiffness’ as Equation 9, where the cross-sectional area and the length are the elements of space we have taken, the latter being the Planck length. Since the Young’s modulus is measured in newtons per square metre, it means the stiffness is given in joules per metre, which we will equate to the ‘spacetime bond energy’ estimated above. By considering Equation 8 it will give us a frequency of order 10⁴³ hertz which is the same order of magnitude as the Planck frequency. Therefore, in order for spacetime stiffness to be higher than the bond energy, the required frequency would be beyond the Planck one, which is not possible.

As a final, and more speculative remark in this reaction, let us mention that since the bond energy has been obtained in a straightforward way out of maximum luminosity, it could be considered as the minimum spacetime bond-breaking energy. As such, it would not be attainable because, in practice, the maximum luminosity of radiated energy is itself a limit only achievable when a highly dynamical astrophysical object reaches exactly its gravitational radius. When such a condition takes place, not even gravitational waves can escape from its inside, and therefore no energy is radiated. Thus, the conclusion is that in situations close to these extreme ones, spacetime would remain ‘unbroken’.

4. A kugelblitz?

Let us recall that our scenario assumes a violent release of energy within a Planck-scale area in a Planck-scale time. We now explore the possibility that such energy is of electromagnetic nature and therefore polarizes the spacetime fabric, leading to the production of electron–positron pairs via the Schwinger effect. The question we seek to address in this last section is whether such an extreme energy concentration could trigger the formation of an event horizon.

Álvarez-Domínguez et al have examined this issue in the context of kugelblitz formation — a hypothetical black hole formed by the gravitational collapse of intense electromagnetic radiation. Their findings indicate that the dissipative effects of the Schwinger mechanism are sufficient to prevent a kugelblitz from forming within length scales ranging from 10⁻²⁹ to 10⁸ metres. Moreover, they highlight that the power required to create it is tens of orders of magnitude beyond what is achievable in any realistic scenario, whether in laboratory conditions or astrophysical environments.

Continuing with the speculative tone of our contribution, let us recall Equation 7 which gives the hypothetical ‘bond energy’ of spacetime, below which a rupture could happen. Álvarez-Domínguez et al describe the dissipation via the Schwinger effect as a sequence of independent non-Markovian dissipation processes of negligibly small duration as compared with the time of formation of the kugelblitz, reasoning that the dissipation time is much smaller than the formation time. The dissipation time is the half-light crossing time for the particle pairs generated inside of a kugelblitz of a given radius. They bound the formation time from below by the time it would take to form the kugelblitz in the absence of dissipation, which is Equation 10; which in our case, by making the radius equal to the Planck length, gives Equation 11. By using Equation 7, we obtain Equation 12. Also in their work, they consider that the dissipation time is bounded from below by the radius divided by the speed of light, arriving to the condition of Equation 13; for our case, with the radius equal to the Planck length, we obtain Equation 14.

Given the value of the bond energy of order 10⁷⁸ joules per square metre and Equation 12, and considering a flux of order 10¹¹⁸ watts per square metre, it gives a formation time of order 10⁻⁴⁰ seconds. Since in our case the dissipation time is the Planck time, of order 10⁻⁴⁴ seconds, the conclusion is that it is not possible that a kugelblitz is formed.

Finally, let us consider, following again Álvarez-Domínguez et al, that in the hypothetical formation of a kugelblitz of a given radius, the electric field would have to get increasingly closer to the one necessary to form a black hole, for which the associated particle pair formation length would be Equation 15. By assuming that pair formation length is the Planck length, an electron mass of order 10⁻³¹ kilograms and an elementary charge of order 10⁻¹⁶ coulombs, the value of the black-hole-forming electric field is of order 10³⁶ joules per coulomb, which is, as expected, orders of magnitude above even the most powerful magnetars known.

5. Conclusions

This brief note offered a new angle for the interpretation of the hypothesized maximum force allowed by General Relativity by taking, as a starting point, the well-known maximum luminosity radiated by a highly dynamic astrophysical system. The novelty of the approach is the use, at face value, of Griffith’s fracture theory, which applies to isotropic and brittle materials. In such a sense, the ‘spacetime fabric’ has been taken, by analogy, as such type of material. Applying then Griffith’s fracture criteria, the limit for the breakage of spacetime at the Planck level involves, in a straightforward way, the hypothesized maximum force. Further considerations directly allow obtaining a hypothetical ‘bonding energy’ between the spacetime building blocks at the Planck level.

Perhaps these considerations could help or motivate the exciting research linked to gravitational waves since they are a consequence of spacetime vibration. Gravitational wave signals are expected over a wide range of frequencies, ranging from 10⁻¹⁷ hertz in the case of ripples in the cosmological background up to the frequencies associated with the formation of neutron stars in supernova explosions. These values are, of course, very well outside of the speculative nature of this article but, in a sort of poetic sense, we never know what surprises might be waiting for us across the vastness of our Universe.

Finally, we have speculated the possible formation of a kugelblitz, concluding that the dissipative effects associated with the vacuum polarization, the Schwinger effect, will prevent such an object from being formed even at Planck scales. Although we once more insist in the pedagogical and speculative nature of this exercise, we also hope that some ideas could lead to perhaps interesting developments.

Data availability statement

There is no data to share. No new data were created or analyzed in this study.