Deriving Einstein's Equations on Causal Sets
From discrete structure to continuum gravity
Extends the capacity-based derivation of Einstein's equations to discrete causal set structures, demonstrating emergence of continuum spacetime.
Feb 27, 2026 · Preprint
Plain English
This paper asks: what if space and time are not smooth, but made of tiny discrete points — can gravity still emerge? The answer is yes.
The problem
We normally think of spacetime as a smooth, continuous fabric. But many physicists suspect that at the smallest possible scale (the Planck length, about 10⁻³⁵ meters), spacetime might actually be made of discrete "atoms" of space. If that is true, can Einstein's gravity still work?
The key idea
A causal set is the most minimal model of discrete spacetime: just a collection of points with a before/after ordering. No distances, no angles, no smooth geometry — only "event A happened before event B." The paper builds a quantum field theory on this bare structure and measures whether gravity emerges.
What the paper does
Using only the causal ordering of random points sprinkled into spacetime, the paper extracts four independent quantities: temperature, entropy, information flow rate, and curvature. All four converge to the values predicted by Einstein's equations as the number of points increases. At 5,000 points, the agreement is exact.
Why it matters
This is evidence that smooth spacetime and gravity are not fundamental — they emerge from something simpler. Even a bag of randomly scattered events with nothing but a notion of "before and after" is enough to produce Einstein's equations. It is a step toward understanding what spacetime actually is at the deepest level.
What could go wrong
The numerical tests are done in 1+1 dimensions (one space, one time), which is simpler than the 3+1 dimensions we live in. Scaling to higher dimensions is computationally expensive and remains to be demonstrated. This is a preprint and has not been peer-reviewed.