WELCOME TO TENSOR ALGEBRA

Not quite the HOTEL CALIFORNIA, the only invariant representation  of a vector is its square, ACCORDING TO DAH RULES.    (DENSITY) =  E_a E^a.   E-sub a times E-super a, Einstein summations. We start, in DKS, with a covariant form.  The two terms are orthogonal. Take E_r and multiply it by g^rr, and you get the CONTRAVARIANT FORM,  E^r. YAH, I have been omitting the sqrt to be complete. . . . . . . .  GAWDDAMMITT, there is no square root. We want energy density.
Here in usual physics FLATLAND, in terms of Lorentz space with only low gravitation, we don’t usually deal with any g_11. If we are hip, we do gravitation as the Schwarzschild metric, where g_11 = 1/ (1 – 2m/r)   . We need only look at the square of any electric field, to know its energy content. THE MATHEMATIC POINT IS, AS SOON AS YOU ALLOW CURVATURE OF SPACETIME, rules are out the window.
That ‘m’ in the Schwarzschild metric is verrrrrrry small in our particle investigations.

Advertisements

22 thoughts on “WELCOME TO TENSOR ALGEBRA

  1. Does this depend on what the goal is? If we want to show something relating to the density of the field lines, we would want the energy Intensity with the square root? The energy density is without the square root. I usually like to consider intensity. Maybe though you have a reason now to look at the energy density?

    In other words what situations would one be interested in density and what situations for intensity?

    Like

    • Simple but not. I NEED ALL THE HELP I CAN GET, to see anything. Be consistent please, maybe one day I’ll be into your actual numbers, yah??? APPLES…………..ORANGES. I hear intelligence in your statement, but do it only for ENERGY DENSITY, please.

      Like

  2. If color rescales to each plot. we must not compare different plots for color !! Perhaps we are coming to an agreed vocabulary, ENERGY DENSITY, and FIELD INTENSITY. I shall try to keep my language straight !!!!! Please tell me when you have the metric terms properly distributed. If what you say of logarithmic dependency is so, then two different plots, one with the sqrt and the other without, should look the same COLORISTICALLY.

    Like

  3. Yes, it took me a while to get to this point, but I can start with an example (albers outer field) where the net effect is the same colors and a shift in the colorbar labeling. We should be OK comparing plots if we look at the labels and if the labels are accurate. The color & labels don’t automatically rescale so we should be OK.

    Good that this exercise is keeping us both honest with our definitions.

    Liked by 1 person

  4. Dude, now we be talkin’ TENSORS, yup. You be awesome. [[ 1/g_11 in this case, with a diagonal metric, is g^11. If we go way down to like Schwarzschild dimensions, the corner matrix term makes things worse. ]]

    Like

  5. It might also be useful to look at the color plots of each component. That is another good idea to include on my new efield web page. Even a blinking animation of the two components and the field.

    Like

  6. There’s actually lots of testing that needs to be done with all the options we’re adding in. I’m attempting to have one basic program do all the bells and whistles that I can toggle on and off. For example with the Kerr metric tensor terms in the field intensity, I’m checking whether the Albers field Tinker Bell point still shows up on the ring edge??

    Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s