"Since Force is a one-form (co-variant vector), is..." solved and explained

High School
Physics
Force, Motion and Energy
Newton's Second Law

Deja Bradshaw

Answered question

2022-10-28

Since Force is a one-form (co-variant vector), is it more accurate to assert that

F = m a^{u} g_{u v}

where

a^{u}

is the acceleration vector, which is contra-variant, and

g_{u v}

is the metric tensor?

Answer & Explanation

Adenomacu

Beginner2022-10-29Added 13 answers

The equation

F = m a^{μ} g_{μ ν}

is notationally unclear. You're right to note that tensor equations have to match types of tensors on both sides, but if we're being really careful about notation, then we have to note exactly what sort of tensor F is. If you mean

F_{ν}

, then it is true that

F_{ν} = m a^{μ} g_{μ ν}

, but it would be equally correct to write

F_{ν} = m a_{ν}

F^{ν} = m a^{ν}

. This is because in Einstein summation notation it is understood that

F^{μ} = g^{μ ν} F_{ν}

and

F_{μ} = g_{μ ν} F^{ν}

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