Efficiently Lowering Tensor Indices: Simplified Equations

In summary: T_{ab} = T_{a}^{b} which is the desired result.In summary, the conversation is about understanding the manipulation of indices and the use of the metric tensor to lower and raise indices. The correct expression is g_{\mu\nu}g_{\mu\nu}T^{\mu\nu} = T g_{\mu\nu} = T_{\mu\nu}. To lower indices, one would multiply by g_{\gamma\mu}g_{\delta\nu} and relabel the indices accordingly. It is important to note that T_{ab}g^{ab} = T_{a}^{a} and T_{ab}g^{bc} = T_{a
Hi
have i got this corresct :
$g_{\mu\nu}g_{\mu\nu}T^{\mu\nu} = T g_{\mu\nu} = T_{\mu\nu}$
and is:
$g_{\mu\nu}g_{\mu\nu}u^{\mu} = g_{\mu\nu} u_{\nu} = u_{\mu}$

No, it should be

gapgqbTpq = Tab

gapvp = va

how would i go about going from $T^{ab}$ to $T_{ab}$ i.e with the same indices that is mainly why i am confused it needs to be the same indices not different ones.

$g_{\gamma \mu }g_{\delta \nu }T^{\mu \nu } = T_{\gamma \delta }$ and then you are free to relabel the indices back to mu and nu. What you wrote is incorrect because you have 2 mu's on the bottom and a mu on the top which isn't how einstein summation works.

ok i think i understand so to lower:
$T^{\mu\nu} = u^{\nu}u^{\mu}$

would one multiply by $g_{\gamma \mu}g_{\delta \nu}$ and relable T as instructed, but how would the u's become:
$u_{\mu}u_{\nu}$

I get the relabelling thing but wouldn't they have three lower indices i.e:
$u_{\mu \gamma \delta} u_{\nu \gamma \delta}$

$g_{\gamma \mu }g_{\delta \nu }T^{\mu \nu } = g_{\gamma \mu }g_{\delta \nu }u^{\mu }u^{\nu }$ so $T_{\gamma \delta } = u_{\gamma }u_{\delta }$ and since these are free indices I can relabel them accordingly that is $\gamma \rightarrow \mu , \delta \rightarrow \nu$ provided I do so on both sides; this gets the desired result.

thanks you have been really helpful one last question, is $T_{ab} g^{ab}= T^{a}_{b}$ if not how can we get there becuase the metric always has two indices?

thanks you have been really helpful one last question, is $T_{ab} g^{ab}= T^{a}_{b}$ if not how can we get there becuase the metric always has two indices?
$T_{ab} g^{ab} = T_{a}^{a}$ because the metric tensor will first raise the lower index b on $T_{ab}$ up to an a and now you just have $T_{a}^{a} = T$ where T is just the trace. Note that here it is arbitrary whether you choose to raise a or b because in the end you end up summing over the same index on the bottom and on the top and you can relabel that to w\e you want; that is $T_{a}^{a} = T_{b}^{b}$ because you end up summing over all components regardless.

thanks

thanks
Yep anytime and to answer your final question you could just use $T_{ab}g^{bc} = T_{a}^{c}$

1. What is the purpose of efficiently lowering tensor indices?

The purpose of efficiently lowering tensor indices is to simplify and streamline calculations in physics and mathematics. Tensor indices represent the dimensions of a tensor, and lowering them involves changing the coordinate system in which the tensor is described. This can make calculations more manageable and intuitive.

2. How does one efficiently lower tensor indices?

Efficiently lowering tensor indices involves using simplified equations that have been developed specifically for this purpose. These equations are based on the properties of tensors and can be applied to any tensor, regardless of its dimensions. By following these equations, the process of lowering tensor indices can be done quickly and accurately.

3. What are the benefits of using simplified equations for lowering tensor indices?

Using simplified equations for lowering tensor indices can save time and effort in calculations. It can also make the results easier to interpret and understand, as they are based on a familiar coordinate system. Additionally, using simplified equations can help avoid errors and ensure accuracy in mathematical and scientific calculations.

4. Are there any limitations to efficiently lowering tensor indices?

Efficiently lowering tensor indices can only be done for tensors that are of a certain rank, or level of dimensions. This means that the equations may not be applicable to all types of tensors. Additionally, the process may become more complex for higher-ranked tensors, and may require more advanced mathematical techniques.

5. What are some real-world applications of efficiently lowering tensor indices?

Efficiently lowering tensor indices is commonly used in fields such as physics, engineering, and computer science. It can be applied to a wide range of problems, such as calculating stress and strain in materials, simulating fluid dynamics, and optimizing algorithms in machine learning. Essentially, anywhere that tensors are used to represent physical or mathematical quantities, the process of efficiently lowering tensor indices can be applied to simplify and improve calculations.

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