--- author: Rowan Brad Quni ORCID: https://orcid.org/0009-0002-4317-5604 tags: [QNFO, AI, quantum, informational universe, IUH, holographic principle] created: 1739878559 modified: 1740658413 --- **Problem:** A particle with a rest mass \( m = 2.0 \, \text{kg} \) is moving at a relativistic momentum \( p = 3.0 \times 10^8 \, \text{kg·m/s} \). 1. **Calculate the total energy \( E \) of the particle using the relativistic energy-momentum relation:** \[ E^2 = (mc^2)^2 + (pc)^2 \] Assume \( c = 3.0 \times 10^8 \, \text{m/s} \). 2. **Discuss why the conservation of energy and momentum in this equation ensures that quantum entanglement cannot transmit information faster than light, as per the no-signaling theorem.** --- **Solution:** 1. **Calculating the total energy:** Substitute the given values into the equation: \[ E^2 = (2.0 \, \text{kg} \cdot (3.0 \times 10^8 \, \text{m/s})^2)^2 + (3.0 \times 10^8 \, \text{kg·m/s} \cdot 3.0 \times 10^8 \, \text{m/s})^2 \] Simplify each term: \[ E^2 = (2.0 \cdot 9.0 \times 10^{16})^2 + (9.0 \times 10^{16})^2 \] \[ E^2 = (1.8 \times 10^{17})^2 + (9.0 \times 10^{16})^2 \] \[ E^2 = 3.24 \times 10^{34} + 8.1 \times 10^{33} = 4.05 \times 10^{34} \, \text{J}^2 \] Taking the square root: \[ E = \sqrt{4.05 \times 10^{34}} \approx 6.36 \times 10^{17} \, \text{J} \] **Final Answer:** \[ E \approx 6.36 \times 10^{17} \, \text{J} \] 2. **Connection to the no-signaling theorem:** The energy-momentum relation ensures that the total energy and momentum of a system are conserved relativistically. In quantum entanglement, while particle states correlate instantaneously, no actual *transfer* of energy or momentum occurs between them. The no-signaling theorem states that these correlations cannot encode measurable information, as doing so would require violating energy-momentum conservation (e.g., altering one particle’s state would necessitate a corresponding change in the other’s energy/momentum, which relativity forbids at superluminal speeds). Thus, the relativistic constraints embedded in \( E^2 = (mc^2)^2 + (pc)^2 \) uphold causality, preventing faster-than-light communication. --- **Key Takeaway:** The energy-momentum relation enforces relativistic limits on physical interactions, ensuring that entanglement correlations remain consistent with causality and the speed of light.