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Component units of the flat metric in length and time coordinates

Physical scope

Unit bookkeeping for the flat model metric under the change of time coordinate from x^0 = ct to x^0 = t: the metric components change from diag(-1,1,1,1) to diag(-c^2,1,1,1) and the component dimensions change accordingly, while the line element ds^2 retains dimension m^2. Illustrates the SI conventions; asserts no measured value of c and no dynamics.

Example

Let $(M, A)$ be the flat Lorentzian four-manifold of The constant Minkowski metric as a smooth time-oriented Lorentzian manifold, whose global chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ is read as length coordinates $x^{0} = c t$ , $(x^{1}, x^{2}, x^{3}) = (x, y, z)$ in metres, with components $A_{a b} = diag (- 1, 1, 1, 1)_{a b}$ , all dimensionless. Introduce the time coordinates $u = (t, x, y, z)$ , so that the coordinate change from $u$ to $x$ is $x^{0} = c u^{0}$ , $x^{i} = u^{i}$ . Then in the chart $u$ the same field $A$ has components

$A_{a b}^{'} = diag (- c^{2}, 1, 1, 1)_{a b},$

the component $A_{00}^{'}$ carries dimension $m^{2} s^{- 2}$ , the components $A_{i j}^{'}$ for $i, j \geq 1$ are dimensionless, and the line element $d s^{2} = A_{a b}^{'} d u^{a} d u^{b}$ again has dimension $m^{2}$ . Thus the same metric has components of different dimensions in the two charts, while its invariant readings are unchanged; this verifies part (b) of SI unit and coordinate conventions of the classical gravitational framework.

Facts & Assumptions

Given: The flat field $A$ with components $diag (- 1, 1, 1, 1)$ in the length chart $x$ on $M = R^{4}$ and the time chart $u$ with $x^{0} = c u^{0}$ , $x^{i} = u^{i}$ for a positive constant $c$ of dimension $m s^{- 1}$ .

[L1]

Under a change of charts $x = x (u)$ the components of a covariant $2$ -tensor field transform as $A_{a b}^{'} (u) = \sum_{c, d} \frac{\partial x^{c}}{\partial u^{a}} (u) \frac{\partial x^{d}}{\partial u^{b}} (u) A_{c d} (x (u))$ (Chart independence and the transformation law of a smooth covariant 2-tensor field).

[L2]

The conventions attach to $c$ the dimension $m s^{- 1}$ ; in length coordinates $x^{0} = c t$ all four coordinates are in metres and the metric components are dimensionless, while in time coordinates $g_{00}$ carries dimension $m^{2} s^{- 2}$ ; the line element $d s^{2} = g_{a b} d x^{a} d x^{b}$ has dimension $m^{2}$ in every chart (SI unit and coordinate conventions of the classical gravitational framework).

[L3]

Constant components in a global chart define a smooth covariant $2$ -tensor field, and the flat field $A$ is the Lorentzian metric of The constant Minkowski metric as a smooth time-oriented Lorentzian manifold (Smooth covariant 2-tensor fields on a smooth manifold).

Verification

technique · direct

1.1L1L3algebra

The Jacobian of the coordinate change is diagonal: $\frac{\partial x^{0}}{\partial u^{0}} = c$ , $\frac{\partial x^{i}}{\partial u^{i}} = 1$ for $i = 1, 2, 3$ , and all off-diagonal entries vanish. Applying [L1] to the diagonal matrix $A = diag (- 1, 1, 1, 1)$ gives $A_{00}^{'} = c^{2} A_{00} = - c^{2}$ , $A_{i j}^{'} = A_{i j} = δ_{i j}$ for $i, j \geq 1$ , and $A_{0 i}^{'} = A_{i 0}^{'} = 0$ ; hence $A^{'} = diag (- c^{2}, 1, 1, 1)$ .

2.1step 1.1L2algebra

By [L2] the factor $c^{2}$ carries dimension $m^{2} s^{- 2}$ , so $A_{00}^{'} = - c^{2}$ carries that dimension, while $A_{11}^{'} = A_{22}^{'} = A_{33}^{'} = 1$ are dimensionless; in particular the components are no longer all dimensionless. In the $u$ chart, $d u^{0} = d t$ carries seconds and $d u^{i}$ metres, so $A_{00}^{'} (d u^{0})^{2}$ and $A_{i j}^{'} d u^{i} d u^{j}$ both carry dimension $m^{2}$ , and so does their sum $d s^{2}$ , in agreement with [L2]; in the $x$ chart the same expression reads $A_{a b} d x^{a} d x^{b}$ with all coordinates in metres.

3.1step 1.1step 2.1L2∎

The two component matrices therefore differ by the factor $c^{2}$ in the time-time entry, and the dimension of a single component depends on the chart while the invariant $d s^{2}$ does not; this is exactly the content of part (b) of SI unit and coordinate conventions of the classical gravitational framework, and the example asserts no numerical value of $c$ .

Component units of the flat metric in length and time coordinates

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources