The previous article explored Newton’s Law of Universal Gravitation, which provided a framework for understanding gravitational interactions. However, certain astronomical observations couldn’t be fully explained by Newtonian mechanics.
In the early 20th century, Albert Einstein re-introduced gravity with his General Theory of Relativity. While Newton’s theory described gravity as a force between masses, Einstein believed that gravity is not a force but a curvature of spacetime caused by mass and energy.
The Limitation of Newtonian Gravity
Newton’s Law of Universal Gravitation works well for most everyday situations and even for the motions of planets in our solar system. However, it has limitations:
Mercury’s Orbit: Observations of Mercury’s orbit showed small deviations from what Newtonian mechanics predicted. It’s perihelion (closest point to the Sun) shifts over time.
High-Speed Objects: Newton’s laws do not account for time dilation and mass increase, which occur when objects move near the speed of light. For instance, GPS satellites need their clocks adjusted because of these relativistic effects.
Massive Objects: Newton’s theory doesn’t explain the curvature of spacetime around massive objects that have extremely strong gravitational fields, such as near a black hole.
General Relativity
In 1905, Einstein published his theory on Special Relativity which talked about how the laws of physics are the same for all non-accelerating observers and that the speed of light is constant regardless of the observer’s motion. Building on these ideas, in 1915, Einstein published his General Theory of Relativity.
Think of spacetime as a flexible fabric that can be warped by mass and energy. Massive objects, like the Sun, cause a curvature in this fabric. Smaller objects, like planets, follow the curves in this fabric, which we perceive as gravitational attraction.
Einstein’s Field Equations (EFE)
He came up with the Einstein’s Field Equations (EFE) to describe how matter and energy influence the curvature of space-time.
Where:
R_μν is the Ricci curvature tensor, which represents the degree to which spacetime is curved by matter. Think of it as a measure of how much spacetime is curved at each point, similar to how we might measure the curvature of a surface like a sphere.
R is the scalar curvature, a singular number that encapsulates the curvature of spacetime at a point.
g_μν is the metric tensor, which defines the geometry of spacetime.
Λ is the cosmological constant, which represents the energy density of empty space or dark energy.
G is the gravitational constant, which determines the strength of gravity.
c is the speed of light.
T_μν is the stress-energy tensor, which describes the distribution and flow of energy and momentum in spacetime.
Einstein's field equations tell us that:
Mass-Energy Curves SpaceTime: Massive objects like stars and planets warp the fabric of spacetime. This curvature tells objects how to move; for instance, in the Solar system, planets follow curved paths around the Sun because of this warping.
Geodesics: Objects move along geodesics, which are the straightest possible paths in curved spacetime. In the absence of other forces, an object will follow a geodesic determined by the curvature of spacetime around it.
Dynamics of the Universe: The equations also describe how the universe evolves over time, including the expansion of the universe and the formation of black holes.
Deriving the Field Equations
Equivalence Principle: Einstein started with the equivalence principle, which states that in a small region of spacetime, the effects of gravity are indistinguishable from acceleration. This principle led him to consider how gravity could be described as a geometric property of spacetime.
Non-Euclidean Geometry: Einstein used the mathematics of non-Euclidean geometry, developed by Bernhard Riemann, to describe curved spacetime. The metric tensor g_μν describes the shape of spacetime and how distances and times are measured.
Stress-Energy Tensor: To link the curvature of spacetime with the matter and energy within it, Einstein used the stress-energy tensor T_μν, which was used in electromagnetism and fluid dynamics.
Ricci Tensor and Scalar Curvature: Einstein needed a mathematical object that described the curvature of spacetime. The Ricci curvature tensor R_μν and the scalar curvature R are derived from the Riemann curvature tensor, which proves how spacetime is curved at every point.
Conservation Laws: The equations had to respect the conservation of energy and momentum, which is naturally handled by the divergence-free property of the stress-energy tensor
\(\nabla_\mu T^{\mu v} = 0\)
Combining these elements, Einstein proposed his field equations:
Where
is the Einstein tensor, which encodes the curvature of spacetime.
The Einstein Tensor
The Einstein tensor G_μν combines the Ricci tensor and scalar curvature to describe how spacetime is curved by matter and energy. The field equations essentially state that this curvature is proportional to the stress-energy tensor, with the proportionality factor involving 8πG/c^4.
Black Holes and Cosmology
General Relativity also predicts the existence of black holes, regions of spacetime where gravity is so strong that not even light can escape. Additionally, it provides the framework for modern cosmology, including the understanding of the Big Bang and the expansion of the universe.
Conclusion
Einstein’s field equations are the crux of General Relativity, describing how matter and energy influence the curvature of spacetime. This theory changed our understanding of gravity from a force acting at a distance to a geometric property of spacetime itself.
They unify the nature of space, time and gravity into a single framework, providing new perceptions of the universe and a deeper understanding to cosmology, black hole physics and gravitational waves.