- #1
ash64449
- 356
- 15
what changes will take place in maxwell's equations if the space-time was curved?
mfb said:The integrated versions can get problematic.
All laws of physics (expressed in their differential form) have basically the same changes. Express them in terms of tensors and replace any derivatives with covariant derivatives. That is it.ash64449 said:what changes will take place in maxwell's equations if the space-time was curved?
DaleSpam said:All laws of physics (expressed in their differential form) have basically the same changes. Express them in terms of tensors and replace any derivatives with covariant derivatives. That is it.
Interesting. I was not aware of your good counter-example. I will have to not say that in the future.WannabeNewton said:Let's be careful here. Maxwell's equations are a perfect example of why naively replacing ##\partial _{a}\rightarrow \nabla_{a}## doesn't always work.
As far as I know, it almost always works but there are examples where it doesn't. It is sort of sad that it doesn't work for the 4-potential form of Maxwell's equations , in the Lorenz gauge, when you think about it, consider they are the most awesome equations known to man (yes I am a Maxwell fanboy).DaleSpam said:Interesting. I was not aware of your good counter-example. I will have to not say that in the future.
DaleSpam said:All laws of physics (expressed in their differential form) have basically the same changes. Express them in terms of tensors and replace any derivatives with covariant derivatives. That is it.
EDIT: this is incorrect, see WannabeNewton's correction below.
WannabeNewton said:Let's be careful here. Maxwell's equations are a perfect example of why naively replacing ##\partial _{a}\rightarrow \nabla_{a}## doesn't always work. Consider the following form of maxwell's equations in flat spacetime: ##\partial ^{a}F_{ab} = -4\pi j_{b},\partial _{[a}F_{bc]} = 0##. If we make the replacement of partials with covariant derivatives in order to make the equations covariant in curved space-time then we simply get ##\nabla ^{a}F_{ab} = -4\pi j_{b},\nabla _{[a}F_{bc]} = 0##. One can show (if you want me to I can show you the calculation right here on this thread) that the inhomogenous Maxwell equations do indeed give us local charge conservation i.e. ##\nabla ^{a}F_{ab} = -4\pi j_{b}\Rightarrow \nabla^{a}j_{a} = 0##.
Consider now the fact that the poincare lemma allows us to claim that locally there exists a one-form ##A_{a}## such that ##F_{ab} = \partial _{a}A_{b} - \partial _{b}A_{a}## (of course ##A_{a}## is just the 4-potential). Writing the inhomogenous Maxwell equations in flat space-time in terms of the 4-potential, after fixing the Lorenz gauge, we simply have ##\partial ^{a}\partial _{a}A_{b} = -4\pi j_{b}##. If we now naively replace the partials with covariant derivatives then we find that ##\nabla^{a}\nabla_{a}A_{b} = -4\pi j_{b}## but this will not give us local charge conservation. By simply replacing partials with covariant derivatives we overlooked the facts that covariant derivatives do NOT commute and that their commutator gives us a curvature term. In fact, the correct form should be ##\nabla^{a}\nabla_{a}A_{b} - R^{c}{}{}_{b}A_{c} = -4\pi j_{b}##. Again you can show with a calculation that this does in fact imply ##\nabla^{a}j_{a} = 0##.
DaleSpam said:Interesting. I was not aware of your good counter-example. I will have to not say that in the future.
WannabeNewton said:As far as I know, it almost always works but there are examples where it doesn't.
Well, I think that's right. Often the same equation can be written in more than one way, e.g. by permuting derivatives, and this will cause the curvature to appear explicitly."No terms explicitly containing the curvature tensor should be added in making the transition from SR to GR" as stated in D'inverno (page 131). It also says this principle is vague and should be used with care, maybe it is referring to cases like the one WB showed.
If we instead look at the differential forms version, ##dF = 0, d^{\star }F = 4\pi^{\star }j## and use Stoke's theorem ##\int_{\Omega } d\omega = \int_{\partial \omega} \omega ## on the second equation we can show e.g. that Gauss's law still holds in GR. See the remarks under part (b) of problem 2 in Chapter 4 of Wald.mfb said:The integrated versions can get problematic.
WannabeNewton said:As far as I know, it almost always works but there are examples where it doesn't. It is sort of sad that it doesn't work for the 4-potential form of Maxwell's equations , in the Lorenz gauge, when you think about it, consider they are the most awesome equations known to man (yes I am a Maxwell fanboy).
DaleSpam said:I guess that you could take the approach that the examples where it doesn't work are not "laws of nature", but in the counter-example you provided I don't like that approach.
WannabeNewton said:Let's be careful here. Maxwell's equations are a perfect example of why naively replacing ##\partial _{a}\rightarrow \nabla_{a}## doesn't always work. Consider the following form of maxwell's equations in flat spacetime: ##\partial ^{a}F_{ab} = -4\pi j_{b},\partial _{[a}F_{bc]} = 0##. If we make the replacement of partials with covariant derivatives in order to make the equations covariant in curved space-time then we simply get ##\nabla ^{a}F_{ab} = -4\pi j_{b},\nabla _{[a}F_{bc]} = 0##. One can show (if you want me to I can show you the calculation right here on this thread) that the inhomogenous Maxwell equations do indeed give us local charge conservation i.e. ##\nabla ^{a}F_{ab} = -4\pi j_{b}\Rightarrow \nabla^{a}j_{a} = 0##.
Consider now the fact that the poincare lemma allows us to claim that locally there exists a one-form ##A_{a}## such that ##F_{ab} = \partial _{a}A_{b} - \partial _{b}A_{a}## (of course ##A_{a}## is just the 4-potential). Writing the inhomogenous Maxwell equations in flat space-time in terms of the 4-potential, after fixing the Lorenz gauge, we simply have ##\partial ^{a}\partial _{a}A_{b} = -4\pi j_{b}##. If we now naively replace the partials with covariant derivatives then we find that ##\nabla^{a}\nabla_{a}A_{b} = -4\pi j_{b}## but this will not give us local charge conservation.
Sure I agree there is nothing remotely deep here. I was just using the EM field as a specific example because making a mistake as simple as assuming ##\nabla^{a}\nabla_{a}## acts on ##A_{b}## like ##\partial ^{a}\partial _{a}## does in the flat space-time case can obstruct one from showing ##\nabla^{a}j_{a} = 0##. With ##\nabla^{a}F_{ab} = -4\pi j_{b}## one manages to doge that bullet whether one knew about the pitfalls of the notation or not and ##\nabla^{a}j_{a} = 0## successful comes out of calculations. Ideally, one could just stick to ##dF = 0, d^{\star }F = 4\pi ^{\star} j## because physical results regarding the EM field can be much more elegantly derived in this form and one can avoid the cumbersome index business to boot.Ben Niehoff said:So the problem is nothing specifically to do with electromagnetic fields at all. It's in making unwarranted assumptions about the Laplace operator (i.e. wave operator). The substitution ##\partial_a \partial^a \rightarrow \nabla_a \nabla^a## only works on scalar fields.
WannabeNewton said:If you wanted to be as natural as possible then going along with dexter, the field equations would invariably come out correctly when you vary the appropriate action for the field theory. If we are talking about a Klein Gordon scalar field then ##\mathcal{L}_{KG} = -\frac{1}{2}\sqrt{-g}(g^{ab}\nabla_{a}\varphi \nabla_{b} \varphi + m^{2}\varphi^{2})## will naturally gives us ##\nabla^{a}\nabla_{a}\varphi - m^2\varphi = 0##. Similarly ##\mathcal{L}_{EM} = -\sqrt{-g}g^{ac}g^{bd}\nabla_{[a}A_{b]}\nabla_{[c}A_{d]}## will give us the correct field equations of electromagnetism on curved space-time. We can simply use a variational principle for the respective field theory on curved space-time and avoid the whole minimal coupling thing.
Thanks for the links. When you mention getting covariant conservation of energy using minimal coupling, do you mean the usual argument of using minimal coupling on ##\partial ^{a}T_{ab} = 0## in flat space-time to get ##\nabla ^{a}T_{ab} = 0## in curved space-time?atyy said:Wouldn't one fail to get covariant conservation of energy without minimal coupling?
atyy said:Wouldn't one fail to get covariant conservation of energy without minimal coupling? I've seen a claim like that in http://arxiv.org/abs/gr-qc/0505128 (Eq 11) and in http://arxiv.org/abs/0704.1733 .
This is also how I have seen it, through Wald (see appendix F). There, the total lagrangian is split up into the lagrangian for gravity and matter fields as ##\mathcal{L} = \mathcal{L}_{G} + \alpha_{M}\mathcal{L}_{M}##. The stress energy tensor is defined through varying the matter action with respect to the metric tensor as ##T_{ab} = -\frac{\alpha_{M}}{8\pi}\frac{1}{\sqrt{-g}}\frac{\delta S_{M}}{\delta g^{ab}}##. Loosely put, using diffeomorphism invariance of ##S_{M}## under a one parameter family of diffeomorphisms and that ##\mathcal{L}_{w}g^{ab} = 2\nabla^{(a}w^{b)}## along the flow of the associated vector field, if the matter field equations are satisfied then one gets ##\nabla^{a}T_{ab} = 0##. The exact details are in appendix F of Wald as noted above. Here I don't see any need for minimal coupling.Ben Niehoff said:The definition I am more familiar with is to split the total Lagrangian into the gravity part and everything else
[tex]\mathcal L_{\text{total}} = \mathcal L_{\text{grav}} + \mathcal L'[/tex]
where ##\mathcal L'## is everything else. Then the EM tensor is defined as
[tex]T_{\mu\nu} = - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}')}{\delta g^{\mu\nu}} = = - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} f(R) \mathcal{L}_m)}{\delta g^{\mu\nu}}[/tex]
for which conservation follows directly as a consequence of the differential Bianchi identity. You can argue about whether the Ricci scalar is "matter", but the point is there should be a conserved tensor of this form.
Bill_K said:Well, I think that's right. Often the same equation can be written in more than one way, e.g. by permuting derivatives, and this will cause the curvature to appear explicitly.
In the example given by WannabeNewton, we're led to a wave equation for Aμ in the form ◻Aμ - Aν;μν = 0, and it's only because we want to impose the Lorenz condition that we need to permute the indices, bringing in the Ricci tensor term.
Maxwell's equations in curved space-time are a set of four partial differential equations that describe the fundamental laws of electromagnetism in the context of Einstein's theory of general relativity. They are used to describe the behavior of electric and magnetic fields in the presence of massive objects and in the curvature of space-time.
We need Maxwell's equations in curved space-time because traditional Maxwell's equations, which were developed in the context of flat space-time, are not applicable in the presence of massive objects or in the curvature of space-time. In order to accurately describe the behavior of electromagnetic fields in these situations, we need to use a modified version of Maxwell's equations that take into account the effects of gravity.
The main difference between Maxwell's equations in curved space-time and traditional Maxwell's equations is that the former incorporates the effects of gravity, while the latter do not. This means that the equations are modified to account for the curvature of space-time and the presence of massive objects, making them more complex and challenging to solve.
Maxwell's equations in curved space-time have numerous applications in astrophysics and cosmology. They are used to study the behavior of electromagnetic fields in the presence of black holes, gravitational waves, and other massive objects. They are also used in the development of models for the early universe and in the study of the formation and evolution of galaxies.
Solving Maxwell's equations in curved space-time requires advanced mathematical techniques and computational methods. One approach is to use numerical simulations, which involve breaking down the equations into smaller parts and solving them iteratively. Another approach is to use analytical methods, which involve finding exact solutions to simplified versions of the equations. Overall, solving these equations is a complex and ongoing area of research in both physics and mathematics.