Spacetime algebra
setting of relativistic physics in geometric algebra
In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) of physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and general relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics".
Spacetime algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.
In comparison to related methods, STA and Dirac algebra are both Clifford Cl1,3(R) algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars.
The STA space–time split is similar to the algebra of physical space (APS, Pauli algebra) approach. APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.
Structure
For any pair of STA vectors,
a
{\displaystyle a}
and
b
{\displaystyle b}
, there is a geometric product
a
b
{\displaystyle ab}
, scalar ('inner') product
a
⋅
b
{\displaystyle a\cdot b}
and exterior ('wedge', 'outer') product
a
∧
b
{\displaystyle a\wedge b}
. The vector product is a sum of a scalar and exterior product:
a
⋅
b
=
a
b
+
b
a
2
=
b
⋅
a
,
a
∧
b
=
a
b
−
b
a
2
=
−
b
∧
a
,
a
b
=
a
⋅
b
+
a
∧
b
.
The public source identifies “Spacetime algebra” as setting of relativistic physics in geometric algebra. This brief keeps that definition visible, then builds a research path around Spacetime, algebra and setting.
Why this record matters
A short description can identify a subject without explaining its stakes. For “Spacetime algebra”, the useful work is to connect “setting of relativistic physics in geometric algebra” to the records capable of establishing context and consequence.
Stable identifiers, scientific names and standards terminology offer the best bridge between this overview and specialist evidence. The source revision retrieved here is dated Sep 13, 2026. The linked authority identifier is Q7572735. None of the 0 selected statements returned an explicit reference.
Scientific names, classifications and consensus can change while older terminology persists in catalogues and historical literature. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Check terminology, classification and the date of the cited evidence. Scientific names and technical consensus can change while older records retain historical value.
- Current terminology
- Classification context
- Finding cited technical literature
Primary datasets, specimen catalogues, standards bodies and the most recent peer-reviewed literature.
Three-step research path
- Establish the record: confirm the title “Spacetime algebra”, its source revision and the description used here.
- Expand the search: follow Spacetime algebra primary sources, Spacetime algebra archive and Spacetime research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Spacetime algebra”?
- Is the terminology current, historical or disputed?
- Has classification or technical consensus changed since the cited source?
Search terms from this dossier
This entry incorporates text from “Spacetime algebra” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.