- #1
fab13
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- TL;DR Summary
- I try to understand a demonstration in special relativity about the continuity equation
deduced from the stress-energy tensor
Hello,
I try to understand how to get the last relation below ##(3)## ( from stress energy tensor in special relativity - Wikipedia ).
I understand how to get equation ##(1)## but I don't grasp how to make appear the gradient operator in the dot product and the divergence operator in the bottom member nabla in equation ##(3)##, i.e ##\partial \nabla \phi_{\alpha}##.
Could anyone could help me to know how to introduce these 2 operators from equation ##(1)## ?
Regards
I try to understand how to get the last relation below ##(3)## ( from stress energy tensor in special relativity - Wikipedia ).
This is to say that the divergence of the tensor in the brackets is ##0##. Indeed, with this, we define the stress-energy tensor:
##
T^{\mu \nu} \equiv \frac{\partial \mathcal{L}}{\partial\left(\partial_{\mu} \phi_{\alpha}\right)} \partial^{\nu} \phi_{\alpha}-g^{\mu \nu} \mathcal{L}\quad(1)
##
By construction it has the property that
##
\partial_{\mu} T^{\mu \nu}=0\quad(2)
##
Note that this divergenceless property of this tensor is equivalent to four continuity equations. That is, fields have at least four sets of quantities that obey the continuity equation.
As an example, it can be seen that ##T_{0}^{0}## is the energy density of the system and that it is thus possible to obtain the Hamiltonian density from the stress-energy tensor.
Indeed, since this is the case, observing that ##\partial_{\mu} T^{\mu 0}=0,## we then have :
##
\frac{\partial \mathcal{H}}{\partial t}+\nabla \cdot\left(\frac{\partial \mathcal{L}}{\partial \nabla \phi_{\alpha}} \dot{\phi}_{\alpha}\right)=0\quad(3)
##
We can then conclude that the terms of ##\frac{\partial \mathcal{L}}{\partial \nabla \phi_{\alpha}} \dot{\phi}_{\alpha}## represent the energy flux density of the system.
I understand how to get equation ##(1)## but I don't grasp how to make appear the gradient operator in the dot product and the divergence operator in the bottom member nabla in equation ##(3)##, i.e ##\partial \nabla \phi_{\alpha}##.
Could anyone could help me to know how to introduce these 2 operators from equation ##(1)## ?
Regards