This paper presents a new regime of relativistic ion acceleration, called the "laser piston" (LP), where an ultra-intense electromagnetic (EM) wave interacts with a thin foil to generate a high-density, ultra-short relativistic ion beam. In this regime, the radiation pressure of the EM wave dominates, allowing efficient conversion of laser energy into ion energy. The ion beam energy per nucleon is proportional to the laser pulse energy, and the process is highly efficient. The LP regime is predicted to be achievable with future Exawatt lasers, offering advantages beyond current Petawatt lasers. The LP regime involves two stages. In the first stage, a relativistic laser pulse irradiates a thin foil, accelerating electrons and creating a charge separation field that accelerates ions. In the second stage, the accelerated foil acts as a relativistic plasma mirror, reflecting the laser pulse and transferring its energy to the ions through radiation pressure. This results in a highly relativistic ion beam with high density and energy. The paper also discusses the suppression of transverse instabilities in the LP regime due to relativistic effects and the stretching of the plasma mirror. 3D particle-in-cell (PIC) simulations confirm the LP regime's feasibility, showing that the ion beam has a high energy, low emittance, and is well collimated. The simulations also show that the ion beam's energy increases with time, following a t^(1/3) dependence. The LP regime has potential applications in high-energy physics, particularly in the development of laser-driven heavy ion colliders. The paper suggests that with future Exawatt lasers, it may be possible to achieve ion beams with energies exceeding 100 GeV per nucleon, suitable for studying quark-gluon plasma. The LP regime is one example of what the authors call "Relativistic Engineering," offering a promising tool for nuclear physics research.This paper presents a new regime of relativistic ion acceleration, called the "laser piston" (LP), where an ultra-intense electromagnetic (EM) wave interacts with a thin foil to generate a high-density, ultra-short relativistic ion beam. In this regime, the radiation pressure of the EM wave dominates, allowing efficient conversion of laser energy into ion energy. The ion beam energy per nucleon is proportional to the laser pulse energy, and the process is highly efficient. The LP regime is predicted to be achievable with future Exawatt lasers, offering advantages beyond current Petawatt lasers. The LP regime involves two stages. In the first stage, a relativistic laser pulse irradiates a thin foil, accelerating electrons and creating a charge separation field that accelerates ions. In the second stage, the accelerated foil acts as a relativistic plasma mirror, reflecting the laser pulse and transferring its energy to the ions through radiation pressure. This results in a highly relativistic ion beam with high density and energy. The paper also discusses the suppression of transverse instabilities in the LP regime due to relativistic effects and the stretching of the plasma mirror. 3D particle-in-cell (PIC) simulations confirm the LP regime's feasibility, showing that the ion beam has a high energy, low emittance, and is well collimated. The simulations also show that the ion beam's energy increases with time, following a t^(1/3) dependence. The LP regime has potential applications in high-energy physics, particularly in the development of laser-driven heavy ion colliders. The paper suggests that with future Exawatt lasers, it may be possible to achieve ion beams with energies exceeding 100 GeV per nucleon, suitable for studying quark-gluon plasma. The LP regime is one example of what the authors call "Relativistic Engineering," offering a promising tool for nuclear physics research.