Raising/Lowering Indices w/ Metric & Tensors: Does Order Matter?

The position of the tensors does not affect the equation. However, the covariant derivative operates on the tensor after it, not the tensors before it. So, while ##g_{uv}V^{v}=V_{u}##, it is not the case for ##V^{v}g_{uv}##.
  • #1
binbagsss
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This is probably a really stupid question , but,
Does it matter whether the metric is after or before the tensor?

My guess is it doesn't because tensors can be positioned in any order, the equation is unchanged.

E.g ##M_{ab}B^{c}T^{m}_{nl}=T^{m}_{nl}M_{ab}B^{c}## right?

However the covariant derivative operates on tensor after it and not the tensors that are before it,

So

##g_{uv}V^{v}=V_{u}##

what about ##V^{v}g_{uv}## does this equal ##V_{u}## or not,

Many thanks
 
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  • #2
binbagsss said:
Does it matter whether the metric is after or before the tensor?
No.
 
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FAQ: Raising/Lowering Indices w/ Metric & Tensors: Does Order Matter?

What is the purpose of raising or lowering indices?

The purpose of raising or lowering indices is to change the form of a tensor while preserving its underlying mathematical properties. This is useful in performing calculations and simplifying equations involving tensors.

Why is the order of indices important when raising or lowering them?

The order of indices is important because it determines the direction in which the tensor is being transformed. Raising or lowering indices in a different order can result in a different tensor altogether.

Can indices be raised or lowered multiple times?

Yes, indices can be raised or lowered multiple times as long as the order in which it is done is consistent. This means that the indices must be raised and lowered in the same order each time to avoid any errors.

What is the difference between raising and lowering indices?

Raising indices involves multiplying the tensor by the metric tensor, while lowering indices involves multiplying the tensor by the inverse of the metric tensor. This results in a change in the form of the tensor, but the underlying mathematical properties remain the same.

Are there any rules or guidelines for raising or lowering indices?

Yes, there are several rules and guidelines for raising or lowering indices, such as the Einstein summation convention, which states that repeated indices in a term are to be summed over, and the metric compatibility condition, which ensures that the metric tensor and the tensor being transformed have the same number of indices and the same type of indices.

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