9.10. Tensor equations

It is not simple to deal with tensor equations because there are many things to worry about simultaneously. Currently xTensor` only works with linear equations where the x tensor is not contracted. The concept of equality (==) has been generalized to include tensor equalities.

From the point of view of Mathematica these two expressions are not related:

In[988]:=

T[a, b] v[-b] == T[a, c] v[-c]

Out[988]=

T_  ^ab v_b^ == T_  ^ac v_c^

However xTensor` recognizes them as equal:

In[989]:=

Simplification[%]

Out[989]=

True

These two are clearly different:

In[990]:=

Simplification[T[a, b] v[-b] == T[-a, c] v[-c]]

Out[990]=

False

Solving equations. The answer is a tensor rule. Options to IndexSolve are actually options to MakeRule:

In[991]:=

rule = IndexSolve[v[a] v[-a] T[b, c, d] == v[b] v[c] v[a] S[-a, d], T[b, c, d], MetricOn→ {b}, ContractMetrics→True]

Out[991]=

In[992]:=

{T[a, b, c], T[-a, b, c], T[a, -b, c], T[-a, -b, c]}/.rule

Out[992]=

{(S_ ( d)^c v_ ^a v_ ^b v_ ^d)/Scalar[v_a^ v_ ^a], (S_ ( d)^c v_a^ v_ ^b v_ ^d)/Scalar[v_a^ v_ ^a], T_ ( b )^(a c), T_ab ^(  c)}

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