A Brief Overview of Lorentz Symmetry and the Structure of Spacetime
Difform motion will, in every case, produce the same effects as gravitation.
Hendrik Lorentz
In 1865, James Clerk Maxwell seminally revealed light’s identity as an electromagnetic wave. Before him, there were scattered postulations about the nature of light, such as Newton’s 17th-century corpuscular theory suggesting that it be composed of massless particles. Such a framework succeeded, for several decades, in describing light as it ricocheted through arrays of lenses and prisms. Although it demonstrably failed in explaining optical phenomena like interference and diffraction (firstly documented by Thomas Young), that were instead explainable through a wave-like paradigm. A diffuse cognizance of this discrepancy percolated throughout the scientific community until Maxwell’s far-reaching reconciliation.
It was then Michelson and Morley who, in 1887, made a discovery that would signal a crisis in physics. In an attempt to detect the then widely-accepted, all-permeating ‘luminiferous aether’ medium that carried Maxwellian light waves, the duo devised an experiment to measure and compare the speed of light in directions perpendicular to the assumed motion of the Earth through the aether. Conceptually, the aether would impede light, reducing its speed. To a perplexed scientific community, their results showed no significant difference between the speed of light. Either the aether did not exist, or the Earth was stationary.
Their publication, On the Relative Motion of the Earth and the Luminiferous Aether, showed the speed of light as independent of the observer’s motion. Their results opened the door for Albert Einstein, who, in the early 20th century, developed a theory of inertial reference frames that preserved the frame-invariant constancy of light’s speed. His 1905 paper On the Electrodynamics of Moving Bodies then introduced the profound view of special relativity to the world.
At its core, special relativity pairs two statements together, with profound implications:
- The laws of physics are the same in all inertial frames of reference. (Lorentz symmetry)
- The speed of light is always the same, regardless of an observer’s relative motion.
This is to say, if you were in a space capsule hurling through the cosmos at 90% light speed, your measurement of the speed of light would be the same as someone on Earth. Amazingly, it is space and time that warp around you, rather than light. In preserving light’s constancy, time must slow down for you, evidenced by a slower-ticking clock relative to the clock on Earth. The twin paradox vividly and succinctly illustrates this concept. If twin A were to travel away from Earth at near light speed, and twin B was to stay behind, twin A would return younger than twin B.
One asks an apparent question that has been the source of much consternation amongst physicists — even more so amongst philosophers: why is the constancy of light’s speed imposed on the laws of nature? Unsatisfyingly, this constraint is deeply rooted in the fabric of reality itself — there is no intuitive answer. In quantum field theory, the speed of light can be understood as a consequence of the properties of the electromagnetic field. Wave-particles of light, photons, are excitations of this field. In this framework, the speed of light arises from the specific structure of the electromagnetic field, which determines how perturbations in the field propagate. This provides no clear answer to the question, though, only a technical redressing of the “how” answer to the “why” question (in fact, “how” is charitable, it is closer to an “if-then”).
Einstein’s special relativity is mind-numbingly fascinating. But the theory is little without the mathematical formalisms introduced in the Lorentz transformation. The transformation is, succinctly put, a set of equations describing how one observer’s measurements of time and space are related to the measurements of another observer traveling at a differing velocity. The Lorentz transformation can be compactly written in matrix notation and as a system of equations:
Which correspond to the general coordinate transform:
The Lorentz transformations can be expressed mathematically in terms of equations that relate the coordinates of an event in one reference frame to those in another. These equations involve a set of parameters, known as the Lorentz factors, that depend on the relative velocity between the frames. The transformations have several important properties, including:
- They are linear and invertible, meaning they can be applied in either direction to transform coordinates between reference frames.
- They preserve the spacetime interval, a quantity that measures the separation between two events in spacetime.
Lorentz symmetry, or Lorentz invariance, refers to the invariance — or changelessness — of the laws of physics under Lorentz transformations. It is one of the two statements paired together in special relativity, and resonates in a myriad of fields today.
The general theory of relativity, formalized by Einstein in 1915, extends the ideas of special relativity to include gravity. In general relativity, from the bends and curves of spacetime determined by the distribution of matter and energy, does gravity emerge. The equations of general relativity are covariant under the whole group of diffeomorphisms (meaning they can be used to describe the same physical situation using different coordinate systems), without changing the results of the calculations. Those diffeomorphisms includes both Lorentz transformations and coordinate transformations that change the spacetime coordinates themselves. This symmetry, known as general covariance, is a more general principle than Lorentz invariance, but is historically rooted in the Lorentz transformation. Likewise is string theory manifestly Lorentz invariant and, broadly, conformally invariant. The latter invariance refers to changes in the scale or conformal factor of the spacetime metric, i.e. it is independent of unit choice or scale to describe spacetime geometry.
More recently, there has been increasing interest in theories violating Lorentz symmetry. Such programs posit that it is a non-fundamental symmetry in nature, and can instead be broken at some level. There is a small, but non-negligible amount of experimental evidence serving as the impetus for such discussions. Namely, observed contradictory interactions between antineutrinos and neutrinos that suggest, hierarchically, the existence of a hypothetical sterile neutrino, laws of physics beyond the Standard Model, or Lorentz symmetry violations. Studies have suggested that anomalous astronomical observations could be evidence of Lorentz-violations, such as cosmic ray flux with a preferred incidence inconsistent with the idea of isotropic cosmic ray sources. Nonetheless, detecting Lorentz symmetry is proposedly testable, and an interesting area of research to follow.
