Varying ##\partial_\lambda\phi\,\partial^\lambda\phi## wrt the metric tensor ##g_{\mu\nu}## in two different ways gives me different results. Obviously I'm doing something wrong. Where am I going wrong?
Method 1: \begin{equation}
(\delta g_{\mu\nu})\,\partial^\mu\phi\,\partial^\nu\phi...
Sorry about the long delay in my reply. I've had a very busy couple of months with deadlines to meet. After having met some of my deadlines, I have returned to reply to your questions as soon as I could manage.
Generally the main purpose of a gauge transformation is to keep the action...
Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then
\begin{align}
g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr}
\end{align}
where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor.
The...
As per the answer given by Demystifier here: https://www.physicsforums.com/threads/are-equations-of-motion-invariant-under-gauge-transformations.1052060/post-6879952 , equations of motion of an action are not necessarily required to be invariant under the gauge transformations of the action.
So...
If a Lagrangian has the fields ##a##, ##b## and ##c## whose equations of motion are denoted by ##E_a, E_b## and ##E_c## respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c
\end{align}
where ##f_1## and ##f_2## are some functions of the fields, if ##E_b## and ##E_c## are...
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
The equations of motion are invariant also under the gauge transformations which are undefined on-shell. ##E_a,\,E_b,## and ##E_c## can be shown to be invariant under such gauge transformations.
I would be happy to come to know of some reasoning which rules out the gauge-transformations that...
Then eq. (2) becomes undefined. Such a gauge-transformation is undefined on-shell. But that does not mean that such a gauge transformation is not possible or not allowed.