Spacetime diagram with light cones illustrating relativistic physics

The Minkowski metric

PHYS 401 · Four-Vectors and Minkowski Spacetime

The Minkowski metric defines spacetime intervals and inner products. This lesson explains metric signatures, invariant dot products, and the difference between Euclidean and spacetime geometry.

Key equations

ds^2=dx^2+dy^2+dz^2ds^2=-c^2dt^2+dx^2+dy^2+dz^2

eta_{mu
u}=egin{pmatrix}-1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}

ds^2=eta_{mu
u}dx^mu dx^
u

A_mu=eta_{mu
u}A^
u

A^mu=(A^0,A^1,A^2,A^3)A_mu=(-A^0,A^1,A^2,A^3)

Acdot B=eta_{mu
u}A^mu B^
u=A_mu B^mu

Acdot B=-A^0B^0+A^1B^1+A^2B^2+A^3B^3ds^2<0ds^2>0ds^2=0-c^2t^2+x^2ds^2=c^2dt^2-dx^2-dy^2-dz^2

Learning objectives

Define the Minkowski interval.
Write the Minkowski metric tensor for $(-,+,+,+)$ signature.
Use the metric to form four-vector inner products.
Classify intervals by the sign of $ds^2$.
Explain the role of metric sign conventions.

Geometry with a different sign

Ordinary Euclidean distance in three-dimensional space is

$ds^2=dx^2+dy^2+dz^2$

All terms have positive signs. Spacetime geometry is different. In special relativity, time contributes with the opposite sign from space. Using the $(-,+,+,+)$ convention,

$ds^2=-c^2dt^2+dx^2+dy^2+dz^2$

This is the Minkowski spacetime interval.

The metric tensor

The Minkowski metric can be written as a matrix:

u}=egin{pmatrix}-1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}$$ Then the interval is $$ds^2=eta_{mu u}dx^mu dx^ u$$ where repeated indices are summed. This is Einstein summation convention. ## Raising and lowering indices The metric converts contravariant components to covariant components: $$A_mu=eta_{mu u}A^ u$$ If $$A^mu=(A^0,A^1,A^2,A^3)$$ then with $(-,+,+,+)$ signature, $$A_mu=(-A^0,A^1,A^2,A^3)$$ The time component changes sign when lowering the index. ## Four-vector inner product The inner product of two four-vectors is $$Acdot B=eta_{mu u}A^mu B^ u=A_mu B^mu$$ In components, $$Acdot B=-A^0B^0+A^1B^1+A^2B^2+A^3B^3$$ This quantity is invariant under Lorentz transformations. ## Classification of vectors A displacement four-vector can be timelike, spacelike, or lightlike depending on the sign of its squared interval. With $(-,+,+,+)$ signature, timelike intervals have $$ds^2<0$$ spacelike intervals have $$ds^2>0$$ and lightlike intervals have $$ds^2=0$$ ## Hyperbolic geometry of boosts Lorentz boosts are analogous to rotations, but with hyperbolic functions instead of circular functions. A boost preserves $$-c^2t^2+x^2$$ rather than preserving $x^2+y^2$. This is why Lorentz transformations involve signs and factors unlike ordinary rotations. ## Metric sign conventions Some authors use the opposite signature $(+,-,-,-)$: $$ds^2=c^2dt^2-dx^2-dy^2-dz^2$$ Both conventions are valid. Equations involving signs must be adjusted consistently. The physics does not depend on the convention. ## The big idea The Minkowski metric defines the geometry of flat spacetime. It gives the invariant interval, determines four-vector inner products, and distinguishes timelike, spacelike, and lightlike directions. Special relativity is not merely about strange effects; it is the geometry of spacetime with a metric that treats time differently from space.

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