# Lorentz invariant mass and length scales

## TL;DR

This paper demonstrates that Lorentz transformations inherently include an invariant mass and length scale, like the Planck scale, ensuring their frame independence without extra principles, and predicts a measurable difference between inertial and gravitational mass.

## Contribution

It shows Lorentz transformations naturally incorporate an invariant scale, impacting quantum gravity spectra and predicting measurable inertial-gravitational mass differences.

## Key findings

- Lorentz transformations admit an invariant Planck-scale.
- Frame independence of geometrical spectra is automatic.
- Predicts a small difference between inertial and gravitational mass.

## Abstract

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such scale is built-in within those transformations, and one does not need to invoke the principle of relativity for their invariance. This automatically ensures the frame-independence of the spectrum of geometrical operators in quantum gravity. Furthermore, we show that the above predicts a small but measurable difference between the inertial and gravitational mass of any object, regardless of its size or whether it is elementary or composite.