The stress-energy tensor explained

The stress-energy tensor.  It sounds scary, and it looks intimidating.  It's a mathematical tool to represent geometry with numbers, and that seldom is easy or clean.

By Maschen, based on File:StressEnergyTensor.svg created by Bamse - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=24940142

The stress-energy tensor is a measure of the potential energy field (AKA spacetime) and how it moves matter. It pushes, it rotates, it stretches and contracts. It’s really all about these red “force” curves.  And it's not all that hard to understand once you have a picture to guide you.

The stress-energy tensor applies to every point in spacetime, here simplified to a single horizontal space axis and vertical time/potential energy axis. Read the indices as “a in the direction of b”. The coordinate scheme is as follows:
0 = time
1 = x axis
2 = y axis
3 = z axis

Energy density is the depth of the red “force” curves. It’s how far the “force” curve (red, inverse square) is below the horizontal axis at any given point. This only requires a single number. (For the particle itself, it’s the green energy line.) Remember, this is the potential energy field. A lack of energy here is the existence of energy in some other form in some other field. Yes, this component is time in the direction of time. Time is, after all, the source and measure of potential energy.

Momentum density is the slope of the red “force” curve. This takes three numbers, one for each direction through space, as modified by time. That’s how you show velocity through spacetime (four-momentum) and the Lorentz factor of relativity. (For the particle itself, it’s the slope of the internal green energy line. In this case, a stationary particle has a slope of 0.)

Shear stress is the rotation of the “force” curve in the three possible planes - xy, xz, yz. It’s curved motion, and is the reason why the Earth is attracted to where the sun is, not where it was 8 minutes ago. I can’t draw well enough to show that. Use your imagination. You’ve seen water forming a whirlpool as it goes down a drain. (See also frame dragging.)

Pressure is the deformation of the pure inverse-square “force” curves. More simply, it represents the stretching or contraction of the curves upon each other as they move through spacetime. If our particle is moving to the right, the curves will be squished together to the right, and stretched out to the left. This is a natural result of the finite rate of propagation of change through the potential energy field. This is better known as the speed of light in a vacuum.

You may have noticed the diagonal symmetry of the stress-energy tensor. You can, for most purposes, ignore the “flux” portion of the chart. Or the other way around. Just pick one, and stick with it.

Gravity and motion

Here we have two different particles in the potential energy field (AKA spacetime). Blue has ten times the mass of red. The force they exert on each other is the inverse square curvature of the field (mass/distance^2), propagated at the speed of causation. How do they move?

Well, assuming a stationary start, they move towards each other. The velocity path they each follow is the sum of the force (acceleration) acting upon them. That propagates as a hyperbola (mass/distance). Here’s what that looks like before the motion starts.

Of course, as they move towards each other, their velocities will increase, and their curves will be gradually compressed in the direction of motion, and stretched out in the opposite direction.

The velocity of a particle is a localized gradient. This gradient has its center at the particle’s mass (which is, of course, subtracted from the potential energy), and is angled down (less potential, more kinetic) in the direction of motion, up in the opposite. The gradient is entirely localized, and does not exist outside the particle’s boundaries (±1 from its center). The combination of this inner energy gradient and the outer curvature is the cause of blue and red shift.

Note that the force acting upon each particle doesn’t care about its mass. The only thing that matters is the curvature of the potential energy field, which is caused by the other particle. But notice how much more the heavier particle curves the field. This causes the lighter particle to accelerate faster than the heavier, and move further before they meet.

Mass curves spacetime. Curvature accelerates mass.

How much does the curvature accelerate the particle? You simply integrate the curvature along the path, and then divide by the time it takes the particle to follow that path. After all, a slower moving particle has more time to be influenced by the curvature than a fast moving one. How do you find the time? Form the particle’s velocity, of course.

The kinetic energy is altered by the path curvature of the potential energy over time. Or, more simply, the momentum gradient is altered by the potential gradient over time.

This is equivalent to the action, which in units is energy multiplied by time (mass * velocity[kinetic] * velocity[potential] * time). Note that in actual use, the mass can often be factored out, because it is both invariant and irrelevant in most cases. (The mass of a satellite doesn’t much affect the motion of the Earth, just as the sun isn’t noticeably perturbed by the orbit of Mars.)

Why does action have the same units as angular momentum? Because what you’re really doing is rotating the angle of the momentum gradient. The principle of least action applies to particles moving in the potential energy field (AKA spacetime) in exactly the same manner as in quantum physics. It is one of the foundations of the equations of general relativity.

Aside 1 - The Heisenberg uncertainty principle shows that the edges of a particle, where the inner kinetic gradient meets the outer potential gradient, are a smooth curve instead of a sharp discontinuity.

Aside 2 - The same principles hold true for the electromagnetic field, except that like charges (those that distort the field’s zero base line in the same direction) repel and opposites attract.

Aside 3 - A stable orbit is that which holds the angle of the momentum gradient constant after one full orbit (which may be different from one rotation). This is a form of quantization.

Copenhagen interpretation delenda est!

Reposted from my substack.

Follow the science!

 The "Science":

“The case against science is straightforward: much of the scientific literature, perhaps half, may simply be untrue. Afflicted by studies with small sample sizes, tiny effects, invalid exploratory analyses, and flagrant conflicts of interest, together with an obsession for pursuing fashionable trends of dubious importance, science has taken a turn towards darkness.”

 - Richard Horton, editor of The Lancet (the premier British medical journal), upon his resignation in 2015

Things have not improved since then.

This is real, published "research" in a "scientific" journal.

How to not water the troops

My son served in a maintenance unit in the 1st Cavalry Division at Fort Cavazos (nee Hood). They participated in weeks-long exercises out in the brush of the maneuver areas regularly. They rotated through training cycles at the Fort Irwin National Training Center (NTC) for at least one month every year. NTC is in the middle of the Mojave desert in California, by the way.

At no point during his three years in that unit did anyone in his command deliver water or food to the soldiers out in the field. Nobody. Not even once. In three years.

Troopers died out in the desert. They died of dehydration and heat stroke.

And nobody in his command cared. Not the officers. Not the senior enlisted. Nobody. Not even once. In three years. It never even crossed their minds that they should.

One summer at NTC, one of the Observer-Controllers (think of them as referees) took pity on my son’s poor unit of maintainers, and ordered a water buffalo delivered to them. It arrived just before their Company Commander made one of his rare checks on them. “Just what I needed!” Then the Captain stripped down, opened it up, and took a bath in their only potable water source.

https://www.armyproperty.com/Equipment-Info/Pictures/Water-Buffalo.jpg

The troops learned a valuable lesson from all this. They learned they had to fend for themselves. They learned to abandon their posts and drive into town to buy food and water. But most of all, they learned that their leaders were both incompetent and uncaring. Nobody cared if they lived or died. Not even once. In three years.

How to not feed the troops

DOGE uncovered the fact that 60% of the money taken from troops’ pay to feed them was used for “other purposes”. That’s not the half of it.

My son joined the Army in the fall of 2020. Not great timing, but what are you going to do? He went to Basic at Fort Sill, Oklahoma, during the winter of 2020-2021. The “dining facilities” (mess halls) were all closed due to a strike and Covid restrictions. That left all the young troops eating nothing but MREs for their entire time in Basic Training. MREs that were stockpiled in sheds. In Oklahoma. In the winter.

These MREs were frozen solid. That’s not too bad, since every MRE comes with a heater pouch. Just add water, and it will warm everything up. Except that the troops were not allowed to use the heaters, because “Somebody might get hurt!”

So they were reduced to bashing their frozen solid MRE meals with rocks. They smashed the packets with rocks until the frozen food was reduced to bite-seized chunks they could chew. Every meal, every day, for three months.

https://www.labroots.com/trending/plants-and-animals/3774/study-chimpanzees-travel-tools-gather-food

And let us not forget the idiotic masking requirements. The idiocy ran so far as to mandate the wearing of masks during physical training (PT). So, when it rained, the troops were essentially forced to waterboard themselves as they exercised and then ran for miles wearing sodden cloth masks. But that’s a different story.

My son was eventually stationed at Fort Cavazos (nee Hood). His battalion lived in brand new barracks. These barracks were so new, he never once had a valid postal address the three years he was there. Their compound included a nice looking mess hall. Which was open for a grand total of almost four weeks (not in a row!) during the three years he was stationed there. This was the mess hall his pay went to fund. If he wanted to eat anywhere else, he had to pay cash.

Were there other dining facilities on the fort? Yes, indeed there were. There were three others. But only one of them was ever open on any given day, with no schedule posted as to which one it might be. And there was no bus or shuttle service to move troops around. So, during their 90 minute lunch break, he could run 3 miles to the nearest DFAC, only to find it was randomly closed that day. And then he could run back to his duty location, still hungry. (Fort Hood has about 28,000 Soldiers scattered over 332 square miles, by the way.)

When the DFAC was open, it often didn’t have any hot food. Lunch was usually a cold table of deli meat and stale bread.  That's if you could get through the long, long lines to get anything at all.  And that's if they didn't simply run out of food before you got any.

No, of course you’re not allowed to cook in the barracks. Somebody might get hurt!

Troops with cars made pocket money as an on-post Uber service.

The US Army cannot or will not feed the troops or deliver their mail to them on permanent bases on US soil.  Why should we have any faith they can prosecute wars overseas?