# Electromagnetic stress-energy formula in wikpedia

• olgerm
In summary: Thanks for spotting that!In summary, the equation for the electromagnetic stress-energy tensor, ##T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,,## can be used to calculate the energy density by replacing ##\mu## and ##\nu## with 0. However, a mistake in the formula on the Wikipedia page results in the incorrect equation of ##T^{0 0}=\sum_{a=0}^D(\eta^{aa}
olgerm
Gold Member
I found equion ##T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,.## from wikipedia page https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor .

it's (0,0) component should be electromagnetic energy density which is ##\frac{E^2+B^2}{2}##.
But by replacing ##\mu## and ##\nu## with 0 I get
energy density=##T^{0 0}=\sum_{a=0}^D(\eta^{aa}*( (F^{0 a})^2+\frac{1}{4} *\sum_{b=0}^D(\eta^{bb}(F^{ab})^2)))=\sum_{a=0}^D(\eta^{aa}*( (F^{0 a})^2+\frac{1}{4} *\sum_{b=0}^D(\eta^{bb}(F^{ab})^2)))=E^2+\frac{B^2}{2}\neq\frac{E^2+B^2}{2}##
Is it my mistake or is the formula wrong?
i am using sign convention where ##\eta^{00}=-1##

Last edited:
mitochan said:
Hi.
I just look through and think ##E## cannot come from ##F^2## s . Is it ##E^2##?
yes, edited it now.

Last edited:
I can't make sense of your formulae, but I don't see any obvious mistake in the Wikipedia article.

vanhees71 said:
I can't make sense of your formulae, but I don't see any obvious mistake in the Wikipedia article.
Can you check whether wikipedia equation equals to ##\frac{E^2+B^2}{2}## if you do not understand my equations?

olgerm said:
Can you check whether wikipedia equation equals to ##\frac{E^2+B^2}{2}## if you do not understand my equations?
It does. If you download Maxima, a free symbolic maths package, the batch file below will generate all the elements of T if you wish to check.
Code:
/* Define the metric */
eta:matrix([-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]);

/* Define the Faraday tensor with upper indices and generate */
/* mixed and lower index versions                            */
uF:matrix([0,-Ex,-Ey,-Ez],[Ex,0,-Bz,By],[Ey,Bz,0,-Bx],[Ez,-By,Bx,0]);
mF:uF.eta;
lF:eta.uF.eta;

/* Contract the Faraday tensor with itself to get an invariant, F */
F:0;
for alpha:1 thru 4 do block (
for beta:1 thru 4 do block (
F:F+uF[alpha,beta]*lF[alpha,beta]
)
);

/* Calculate the stress-energy tensor */
T:-(1/4)*eta*F;
for mu:1 thru 4 do block (
for nu:1 thru 4 do block (
for alpha:1 thru 4 do block (
T[mu,nu]:T[mu,nu]+uF[mu,alpha]*mF[nu,alpha]
)
)
);

/* Display the t-t element of T */
ratsimp(T[1,1]);
Edit: note that the eta after the uF in both lines 7 and 8 should strictly be transpose(eta), but since the metric is always symmetric this has no effect.

Last edited:
olgerm
Ibix said:
It does. If you download Maxima, a free symbolic maths package, the batch file below will generate all the elements of T if you wish to check.
It seams to me that according to that script T[1,1]=(Bx^2+By^2+Bz^2+2*Ex^2+2*Ey^2+2*Ez^2)/2=##E^2+\frac{B ^2}{2}##, but that should be ##\frac{E^2+B^2}{2}##. if the formula were correct.
Does it use the same equation that is on my 1. post to generate the matrix?

olgerm said:
It seams to me that according to that script T[1,1]=(Bx^2+By^2+Bz^2+2*Ex^2+2*Ey^2+2*Ez^2)/2=##E^2+\frac{B ^2}{2}##, but that should be ##\frac{E^2+B^2}{2}##. if the formula were correct.
That's because I somehow posted an incorrect version that calculated ##F_{\mu\nu}## as ##\eta_{\mu\rho}F^{\rho\nu}## instead of ##\eta_{\mu\rho}\eta_{\nu\sigma}F^{\rho\sigma}## (i.e. I calculated the mixed tensor and used it as the lower tensor). Not sure how I managed that. I've corrected the script above (line 8 is the only change) and it gets the correct answer now.

Last edited:
olgerm

## 1. What is the electromagnetic stress-energy formula in Wikipedia?

The electromagnetic stress-energy formula in Wikipedia is a mathematical representation of the relationship between electromagnetic fields and energy. It is also known as the electromagnetic energy-momentum tensor and is used to describe the distribution of energy and momentum in electromagnetic fields.

## 2. How is the electromagnetic stress-energy formula derived?

The electromagnetic stress-energy formula is derived from Maxwell's equations, which describe the behavior of electric and magnetic fields. By combining these equations with the principles of special relativity, the formula can be derived to represent the energy and momentum of electromagnetic fields.

## 3. What is the significance of the electromagnetic stress-energy formula?

The electromagnetic stress-energy formula is significant because it allows for the calculation of the energy and momentum of electromagnetic fields, which are fundamental to many physical phenomena. It also provides a way to understand the relationship between energy and momentum in electromagnetic fields.

## 4. How is the electromagnetic stress-energy formula used in practice?

The electromagnetic stress-energy formula is used in various fields of physics, such as electromagnetism, quantum field theory, and general relativity. It is also used in practical applications, such as in the design of electronic devices and in the study of electromagnetic radiation.

## 5. Are there any limitations to the electromagnetic stress-energy formula?

Like any scientific formula, the electromagnetic stress-energy formula has its limitations. It does not account for certain phenomena, such as quantum effects, and is only applicable in the context of special relativity. It also assumes a vacuum environment, which may not always be the case in real-world situations.

• Electromagnetism
Replies
2
Views
238
• Advanced Physics Homework Help
Replies
1
Views
704
• Special and General Relativity
Replies
3
Views
1K
• Advanced Physics Homework Help
Replies
2
Views
447
• Electromagnetism
Replies
5
Views
925
• Calculus
Replies
1
Views
1K
• Calculus
Replies
5
Views
880
• Special and General Relativity
Replies
2
Views
1K
• Electromagnetism
Replies
6
Views
1K
• Special and General Relativity
Replies
5
Views
499