Moment tensor

Builds on Seismic moment Symmetric matrix Green's function Torque Trace of a matrix

Definition

Let an elastic medium have elastodynamic Green's function $G_{np}(x, t; \xi, \tau)$ . It is the $n$ -th component of displacement at $(x, t)$ due to a unit impulsive force in direction $p$ at $(\xi, \tau)$ . A point source at $\xi$ has moment tensor $M_{pq}(t)$ , a real symmetric $3 \times 3$ matrix for each time $t$ , if the displacement it produces is $u_n(x, t) = \sum_{p,q} \int M_{pq}(\tau)\, \frac{\partial G_{np}}{\partial \xi_q}(x, t; \xi, \tau)\, d\tau.$ The entry $M_{pq}$ is the strength of a couple of forces in direction $p$ separated in direction $q$ .

Remarks

Symmetry, $M_{pq} = M_{qp}$ , says the source exerts no net torque; an internal source cannot change the angular momentum of the body. Example: slip in the unit direction $d$ on a small fault with unit normal $n \perp d$ and seismic moment $M_0$ has $M = M_0 (n d^T + d n^T)$ . This is symmetric, and its trace is $2 M_0\, n \cdot d = 0$ , so the source causes no volume change. An explosion has $M = M_0 I$ , with trace $3M_0 \ne 0$ . Non-example: a single couple, $M = M_0\, e_1 e_2^T$ , is not symmetric, so it is not a moment tensor; it would exert a net torque. Every moment tensor splits as an isotropic part $\frac13 (\tr M) I$ plus a deviatoric part of trace zero.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Green's function concept
Matrix definition
Seismic moment concept
Symmetric matrix concept
Torque concept
Square matrix definition
Trace of a matrix definition
Moment tensor definition

Tag	`QBZC7ZJTWP`
URI	`m2://QBZC7ZJTWP`
Kind	definition
Writing	W0 stub
MSC	86A15, 74J05, 86A22
Defines	Moment tensor
Uses	Seismic moment, Symmetric matrix, Green's function, Torque, Trace of a matrix
Changed	3 November 2026

Moment tensor

Definition

Remarks

Reading path

Further reading

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