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Where do the unit labels go on tensors?

This discussion is continued from this thread https://www.physicsforums.com/showthread.php?t=321298"

Greg, I've been trying to make sense of things in mixed units. Taking the dot product as you've done is a good test, Dale.

x·x = x_{μ}x^{μ}

If x^{μ}has units (T, L, L, L) then x_{μ}has units (1/T, 1/L, 1/L, 1/L)

The scalar product would be unitless.

The metric has interesting units. So that x_{ν}= η_{μν}x^{μ}, works out, the metric would have units

[tex]

\left[ \begin {array}{c}

{\frac{1}{T^2} \; \frac{1}{LT} \; \frac{1}{LT} \; \frac{1}{LT}} \\

{\frac{1}{LT} \; \frac{1}{L^2} \; \frac{1}{LT} \; \frac{1}{LT}} \\

{\frac{1}{LT} \; \frac{1}{LT} \; \frac{1}{L^2} \; \frac{1}{LT}} \\

{\frac{1}{LT} \; \frac{1}{LT} \; \frac{1}{LT} \; \frac{1}{L^2}}

\end{array} \right]

[/tex]

I don't know if it makes sense in terms of the full tensor.

[tex] \eta = \eta_{\mu \nu}\: dx^{\mu} \otimes dx^{\nu} [/tex]

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω . . . . . Γ Δ Θ Λ Ξ Π Σ Φ Ψ Ω

∂ ∫ ∏ ∑ . . . . . ← → ↓ ↑ ↔ . . . . . ± − · × ÷ √ . . . . . ¼ ½ ¾ ⅛ ⅜ ⅝ ⅞

∞ ° ² ³ ⁿ Å . . . . . ~ ≈ ≠ ≡ ≤ ≥ « » . . . . . † ‼

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# Tesnsors and Units Consistancy

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