# Counterexamples to the Maximum Force Conjecture

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- Strong form: $f\le \frac{1}{4}$.
- Weak form: $f=\mathcal{O}\left(1\right)$.

## 2. Spherical Symmetry

## 3. Perfect Fluid Spheres

#### 3.1. Generalities

#### 3.2. Schwarzschild’s Constant Density Star

#### 3.2.1. Radial Force

#### 3.2.2. Hemspherical Force

#### 3.2.3. DEC

#### 3.2.4. Summary (Schwarzschild Interior)

#### 3.3. Tolman IV Solution

#### 3.3.1. Radial Force

#### 3.3.2. Hemispherical Force

#### 3.3.3. DEC

#### 3.3.4. Summary (Tolman IV)

#### 3.4. Buchdahl–Land Spacetime: $\rho ={\rho}_{s}+p$

#### 3.4.1. Radial Force

#### 3.4.2. Hemispherical Force

#### 3.4.3. DEC

#### 3.4.4. Summary (Buchdahl–Land)

#### 3.5. Scaling Solution

#### 3.5.1. Radial Force

#### 3.5.2. Hemispherical Force

#### 3.5.3. Summary (Scaling Spacetime Geometry)

#### 3.6. TOV Equation

#### 3.6.1. Radial Force

#### 3.6.2. Hemispherical Force

#### 3.6.3. Summary (TOV)

## 4. Discussion

“It is not true that I proposed the formula ${c}^{5}/G$ as a luminosity limit for anything. I make no such claim. Perhaps this notion arose from a paper that I wrote in 1962 with the title, “Gravitational Machines”, published as Chapter 12 in the book, “Interstellar Communication” edited by Alastair Cameron, [New York, Benjamin, 1963]. Equation (11) in that paper is the well-known formula $128{V}^{10}/5G{c}^{5}$ for the power in gravitational waves emitted by a binary star with two equal masses moving in a circular orbit with velocity V. As V approaches its upper limit c, this gravitational power approaches the upper limit $128{c}^{5}/5G$. The remarkable thing about this upper limit is that it is independent of the masses of the stars. It may be of some relevance to the theory of gamma-ray bursts”.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TOV | Tolman–Oppenheimer–Volkoff |

DEC | Dominant energy condition |

SEC | Strong energy condition |

WEC | Weak energy condition |

NEC | Null energy condition |

## Note

1 | Awarded 4th prize in the 1962 Gravity Research Foundation essay contest: (https://www.gravityresearcrthfoundation.org/year, accessed on 27 October 2021) |

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**Figure 1.**Radial force ${F}_{r}(\chi ,y)$ for the interior Schwarzschild solution. Note: ${F}_{r}(\chi ,y)$ is bounded above by 2 in the region of interest $y\in [0,1]$, $\chi \in [0,8/9)$.

**Figure 4.**Radial force ${F}_{r}(\chi ,y)$ for the interior Schwarzschild solution in region 1 $\left(0\le \chi \le \frac{3}{4},\phantom{\rule{1.em}{0ex}}0\le y\le 1\right)$ where the DEC is satisfied.

**Figure 5.**Radial force ${F}_{r}(\chi ,y)$ for the interior Schwarzschild solution in region 2 $\left(\frac{3}{4}<\chi \le \frac{8}{9},\phantom{\rule{1.em}{0ex}}4-\frac{3}{\chi}\le y\le 1\right)$ where the DEC is satisfied.

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