7
votes
5answers
466 views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
1
vote
0answers
42 views

Allowed transformations in General Relativity [duplicate]

So in Special Relativity we have: $$ \Lambda \eta \Lambda^T=\eta $$ Is there an analagous formula for the metric in General Relativity?
6
votes
2answers
337 views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
3
votes
2answers
180 views

The signature of the metric and the definition of the electromagnetic tensor

I've read the definition of the electromagnetic field tensor to be ...
1
vote
1answer
30 views

Flat space current conservation sign confusion

It is said that in Minkowski spacetime, the current conservation law for the number current $N^\mu$ where $N^0$ is the number density and $N^i, i=1,2,3$ is the particle flux in the $x^i $ direction, ...
2
votes
2answers
128 views

Distinguish between Past and Future

When writing the metric in Minkowski space, how can we distinguish between the past and the future? I understand the answer after drawing the light cone but I want to know how we get that by just ...
2
votes
1answer
187 views

Lightcone coordinates

The Light cone coordinates are defined as $$x^+ = x^0 + x^3$$ $$x^- = x^0 - x^3$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach in his A First ...
3
votes
3answers
590 views

How the Lorentz transformation affects the metric tensor?

After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure: and in such coordinate system, according to this Wikipedia article, the metric ...
1
vote
2answers
149 views

Sign convention for basic Dirac equation

The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0 $$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0 $$ in a (+,---) metric right?
0
votes
1answer
227 views

Lorentz transformation problem

In the equation (1.18) they omitted the translation vector $a^\mu$, but why?
3
votes
2answers
395 views

Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
3
votes
1answer
248 views

Can the overall sign of the Minkowski metric be changed?

If we take the Minkowski metric, $\eta_{\mu\nu}=(1,-1,-1,-1)$, instead of the usual $(-1,1,1,1)$, does this change the form of the Lorentz Transform? I think the standard Lorentz Transform looks like: ...
4
votes
1answer
140 views

Intervals as infinitesimals of same order (Landau & Lifshitz)

I don't understand the following statement in Landau & Lifshitz, Classical Theory of Fields, p.5: $ds$ and $ds'$ are infinitesimals of same order. [...] It follows that $ds^2$ and $ds'^2$ must ...
1
vote
2answers
128 views

What should I call an n>4 dimensional Minkowski metric?

I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be ...