General Covariance in Theoretical Physics
In the realm of theoretical physics, general covariance—also referred to as diffeomorphism covariance or general invariance—is the principle that the form of physical laws remains unchanged under arbitrary differentiable coordinate transformations. At its core, this concept posits that coordinates are not inherent properties of nature but are merely human-made tools used to describe it. Consequently, the choice of a coordinate system should have no influence on the formulation of fundamental physical laws.
While this principle is a cornerstone of general relativity, it is not expected to apply to all theories. For instance, matter fields that exist independently of their background typically exhibit different equations of motion in curved space compared to flat space.
Key Facts
- Definition: The invariance of physical laws regardless of the coordinate system used.
- Mathematical Tool: Generally covariant laws are typically expressed using tensor fields.
- Core Philosophy: Coordinates are artifices, not a priori natural entities.
- Primary Application: It is the foundational framework for Albert Einstein's general theory of relativity.
- Modern View: It is interpreted as the Lie group GL 4 (R) acting as a fundamental external symmetry of the world.
The Evolution of Covariance: From Special to General Relativity
Albert Einstein initially proposed the principle of covariance for his special theory of relativity. However, special relativity was limited to spacetime coordinate systems related by uniform inertial motion—meaning relative motion in a straight line without acceleration. This is known as global Lorentz covariance.
Einstein eventually realized that the principle of relativity must also encompass accelerated motions. To achieve this, he utilized tensor calculus to extend the theory from global Lorentz covariance to local Lorentz covariance, which applies to all frames of reference. This transition led to the development of the general theory of relativity.
In this framework, the local reduction of the metric tensor (a mathematical object describing the geometry of spacetime) to the Minkowski metric tensor represents free-falling, or geodesic, motion. This mathematical relationship allows the theory to encompass the phenomenon of gravitation.
Unified Field Theories and Geometric Interpretation
Following the success of general relativity, much of the effort in classical unified field theories focused on extending general covariance to other physical phenomena. Scientists attempted to interpret electromagnetism and other forces as purely geometric objects within the spacetime continuum, treating them as extensions of the geometric framework established by Einstein.
Historically, there has been some confusion regarding the terminology. In 1917, the mathematical tools available were not yet refined enough to distinguish between the demand for "no prior geometry" and the demand for a coordinate-independent formulation of physics. Einstein used the term "general covariance" to describe both, which led to decades of theoretical ambiguity.
| Type of Covariance | Scope of Application | Associated Theory |
|---|---|---|
| Global Lorentz Covariance | Uniform inertial motion (no acceleration) | Special Relativity |
| Local Lorentz Covariance | All frames, including accelerated motion | General Relativity |
| General Covariance | Arbitrary differentiable coordinate transformations | General Relativity / Unified Field Theories |
Frequently Asked Questions
What is the main purpose of general covariance?
The main purpose is to ensure that the fundamental laws of physics are independent of the coordinate system used to describe them, reflecting the idea that coordinates are human constructs rather than physical realities.
How does general covariance differ from special relativity?
Special relativity relies on global Lorentz covariance, which only applies to inertial frames moving at constant velocities. General covariance extends this to all coordinate systems, including those undergoing acceleration.
What role do tensor fields play in this concept?
Tensor fields are the mathematical language used to express generally covariant laws. Because tensors transform predictably under coordinate changes, equations written in terms of tensors maintain the same mathematical form across all systems.
Is general covariance applicable to all physical theories?
No. While it is essential for fundamental theories like general relativity, it is rarely found in theories where matter fields are treated as existing independently of the background spacetime.
What is the modern interpretation of the principle?
Modern physics interprets the principle of general covariance as the Lie group GL 4 (R) serving as a fundamental "external" symmetry of the universe, alongside various "internal" symmetries based on compact groups.