# Energy-momentum tensor: metric tensor or kronecker tensor appearing?

• Ameno
In summary, the question is about the correct definition of the energy-momentum tensor in terms of the Lagrangian density, with two different possible definitions provided. The conversation then discusses the convention of using the mixed version of the metric tensor and the differences between using the Kronecker delta and the metric tensor. The article linked provides clarification on this matter.
Ameno
Hi

This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions:
$${T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu \phi - {\delta^\mu}_\nu \mathcal{L}$$
$${T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu \phi - {g^\mu}_\nu \mathcal{L}$$
Which of the two is correct? Is this somehow a matter of convention or something like that? I have seen both more than once.

Thanks, this makes sense. Am I right to say that
$$T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - \delta^{\mu\nu} \mathcal{L}$$
and
$$T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - g^{\mu\nu} \mathcal{L}$$
are the same only if one makes the convention that $${\delta^\mu}_\nu$$ is no longer just a symbol for the kronecker delta but a tensor, namely $${\delta^a}_b = g^{a c} g_{c b}$$ and that then, $$\delta^{ab} = {\delta^a}_b g^{bc} = g^{ab}$$ but that the two are not the same if $$\delta^{\mu \nu}$$ is understood as the kronecker delta?

If I see things correctly, one has to look carefully if an appearing delta is just a symbol for the kronecker delta in components or really a (raised or lowered) version of the metric tensor. This is slightly confusing.

## 1. What is the energy-momentum tensor?

The energy-momentum tensor is a mathematical object used in physics to describe the distribution of energy and momentum in a system. It is a rank-2 tensor, meaning it has two indices, and is used to represent the energy and momentum at each point in spacetime.

## 2. Is the energy-momentum tensor a metric tensor?

No, the energy-momentum tensor is not a metric tensor. Metric tensors are used to define the geometry of spacetime, while the energy-momentum tensor describes the energy and momentum within that geometry.

## 3. How is the energy-momentum tensor related to the stress-energy tensor?

The stress-energy tensor is another name for the energy-momentum tensor, and the terms are often used interchangeably. However, the stress-energy tensor may also refer to a specific form of the energy-momentum tensor that describes the stress and energy flux in a fluid or solid medium.

## 4. Can the energy-momentum tensor be derived from the metric tensor?

No, the energy-momentum tensor cannot be derived from the metric tensor. While the metric tensor is used to define the geometry of spacetime, the energy-momentum tensor is a separate physical quantity that describes the distribution of energy and momentum within that geometry.

## 5. What is the significance of the Kronecker delta in the energy-momentum tensor?

The Kronecker delta is often used in the energy-momentum tensor because it represents the identity matrix, which is necessary for the tensor to have the correct properties. However, the Kronecker delta does not have any physical significance in the context of the energy-momentum tensor.

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