Equation 1 · How Much of Gravitationally Induced Entanglement Is in the Eye of the Frame
What does this equation mean?
Imagine each mass taking a left or right path. Pick branch i for mass A and branch j for mass B. Their separation sets a gravitational interaction energy; over time that energy rotates the complex amplitude of that joint path by the angle shown here. Different paths can acquire different angles.
Four paths, four phases
The two masses each have L and R spatial branches. Their joint state has four branch pairs: LL, LR, RL, and RR. Each pair has a potentially different distance ; consequently, each acquires a potentially different phase φ_{ij}. The phase is attached to the amplitude of a possible joint path, not to a classical trajectory that has already been observed.
Read it piece by piece
Phase of branch pair i,j
The dimensionless phase angle, in radians, acquired by the joint path in which mass A uses branch i and mass B uses branch j. Its individual value depends on phase convention; differences between branch-pair phases can affect interference and entanglement.
Explore this idea →Branch labels
Each label is L or R. The first chooses mass A’s path and the second chooses mass B’s path, giving LL, LR, RL, and RR.
Explore this idea →Newtonian gravitational constant
The constant that sets the strength of Newtonian gravity. Multiplying G by both masses and dividing by their separation gives the magnitude of their interaction energy.
Mass of A
The mass of the first object, measured in kilograms. The phase grows in proportion to this mass while the other quantities are fixed.
Mass of B
The mass of the second object, measured in kilograms. The phase grows in proportion to the product .
Interaction time
How long the two branches maintain the gravitational interaction. Doubling this duration doubles the angle in this idealized constant-separation model.
Distance for this branch pair
The separation between mass A on branch i and mass B on branch j. A smaller separation gives a larger gravitational interaction energy and therefore a larger accumulated phase.
Reduced Planck constant
The quantum constant that converts energy multiplied by time (action) into a phase angle. Dividing the gravitational energy-time product by ħ leaves a dimensionless number of radians.
How to interpret it
The Newtonian potential energy for a branch pair is −G /. Quantum time evolution turns energy multiplied by time into phase: −t/ħ gives the positive angle written here under this sign convention. The energy-time product has units of joule-seconds, matching ħ, so the result is an angle rather than a force or a distance. Only relative phases enter observable interference; one branch phase alone is not an entanglement measure.
Try the four branch phases
This is an idealized geometry: mass A and mass B each have a left and right branch. Matching branches are separated by a gap d; crossed branches are farther apart by a horizontal offset Δx. Change the values to see which phase difference survives.
Illustrative symmetric geometry. The phase and concurrence calculations assume coherent, stationary branches and no other interactions. The article’s physical experiment has additional constraints.
Why mass, distance, and time matter
The interaction-energy magnitude is G /. The masses multiply, the distance appears in the denominator, and time accumulates the effect. If all four branch distances were equal, all four phases would be equal and this gravitational interaction would supply no relative phase between the paths.
Conditions for this formula
This is the weak-field, Newtonian, fixed-separation approximation for coherent, localized branches. Motion during the interaction, other forces, environmental decoherence, and a more complete gravitational treatment require additional analysis. It is an idealized phase model, not by itself a prediction of a completed experiment.
Sources and further reading
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