G_μν = 8πT_μν — the most beautiful equation in physics
Einstein's field equations G_μν = 8πG T_μν/c⁴ relate the curvature of spacetime (left side) to the distribution of matter and energy (right side). They are ten coupled, nonlinear partial differential equations — extraordinarily difficult to solve, yet elegant in their compression of all gravitational physics into a single tensor equation.
This page explores the stress-energy tensor and the real-world applications that make GR not just beautiful but essential for modern technology and astronomy.
The stress-energy tensor T_μν is a 4×4 symmetric matrix that describes the density and flux of energy and momentum. Its components include energy density, momentum density, pressure, and shear stress. It is the "source" of gravity in Einstein's equations.
Click any cell in the matrix to see what that component of T_\u03bc\u03bd represents physically.
GPS satellites orbit at 20,200 km altitude and 14,000 km/h. Special relativity says their clocks run slower (by 7 μs/day) due to their speed. General relativity says they run faster (by 45 μs/day) due to weaker gravity. The net effect: satellite clocks gain 38 μs/day. Without correction, GPS would drift by ~10 km/day.
GPS satellites experience +45.85 us/day (gravity) and -7.21 us/day (velocity), net +38.64 us/day correction needed.
Key insight: Your phone's GPS is a relativistic instrument. Every position fix relies on Einstein's equations being correct to parts per billion.
Clocks at different altitudes tick at different rates. A clock at sea level ticks slower than one on a mountaintop. The effect is tiny — about 1 part in 10¹⁶ per meter of altitude — but modern atomic clocks can measure it across a height difference of just 1 cm.
Clocks at higher altitudes experience weaker gravity and tick faster. GPS orbit is at ~20,200 km.
A radar signal sent past a massive object takes slightly longer than expected — not because the path is longer, but because time itself runs slower near the mass. Irwin Shapiro predicted this "fourth test of GR" in 1964, and it was confirmed by bouncing radar off Mercury and Venus as they passed behind the Sun.
Radar signals passing near the Sun are delayed. Closer approach = larger delay. This was the fourth classical test of GR.