Grok's Answer to 'Why Don't Definitions for Energy Work in Relativistic Space-time'

 Grok provided a reasonable answer...

"In non-relativistic physics, energy is beautifully simple:
Total energy = kinetic energy + potential energy + internal energy, and it works perfectly because:

• Time is absolute (same for everyone).
• Space is Euclidean and absolute.
• Mass is invariant.
• You can cleanly separate "kinetic" from "rest" contributions.

In special relativity, none of these assumptions hold, and that’s why the old definitions of energy break down or become ambiguous/meaningless. Here’s why each common non-relativistic notion of energy fails:1. Kinetic Energy as (1/2)mv²Fails spectacularly.
At high speeds, momentum is p = γmv (where γ = 1/√(1−v²/c²)), not mv.
The work–energy theorem still holds, but when you integrate F·dx = dp·v along a path, you get:Total energy E = γ m c²
Kinetic energy K = (γ − 1) m c²The classical (1/2)mv² is only the low-speed approximation (Taylor expansion of γ − 1 ≈ v²/2c²).2. Rest Energy (or "mass energy") wasn’t even a conceptClassically, energy can be zero when an object is at rest.
In relativity, even a particle at rest (in some frame) has energy E = m c².
You cannot separate "rest mass energy" from "kinetic energy" in a frame-independent way — the split depends on who is looking.3. Potential Energy becomes frame-dependent and problematicExample: Two charges at rest in one frame have electrostatic potential energy.
Boost to another frame → they’re moving → there are magnetic fields → the energy is now partly in the fields, partly kinetic, etc.
The division between "potential energy in fields" and "kinetic energy of particles" is not Lorentz-invariant. Different observers disagree on how much energy is potential vs kinetic vs field energy.4. Total energy is not a scalar in the old senseIn relativity, energy is the time component of the four-momentum vector: P^μ = (E/c, p_x, p_y, p_z)The magnitude P^μ P_μ = (m c)² is invariant — that’s the rest mass.
But the energy E itself transforms under Lorentz boosts, just like time and space do. So "total energy" is observer-dependent.5. Conservation laws require the full energy-momentum tensorFor continuous systems (fields, fluids), the old idea of "total energy = ∫ kinetic + potential + internal" fails completely.
You need the stress-energy tensor T^{μν}, whose component T^{00} is energy density, but the total 4-momentum of a system is only well-defined if the system is isolated or you specify a frame carefully.Summary: Why ordinary definitions don’t work

Concept	Non-relativistic assumption	Why it fails in SR
Kinetic energy = ½mv²	Linear momentum p = mv	Momentum is γmv; wrong low-speed limit only
Rest energy = 0	Mass has no energy when v=0	E = mc² even at rest
Potential energy	Frame-independent	Depends on observer (mixes with kinetic/field)
Total energy	Scalar, same for all observers	Part of 4-vector; different in different frames
Energy conservation	Just conserve scalar E	Must conserve full 4-momentum

The relativistic fixWe stop trying to split energy into "kinds" in a universal way and instead define:

• Four-momentum P^μ = (E/c, p)
• Invariant mass m²c² = E²/c² − p² (the true, frame-independent "mass")
• Total energy in a given frame E = √(p²c² + m²c⁴)

That’s the only consistent, frame-independent way to talk about energy in special relativity. All the comfortable classical categories dissolve — just like space and time themselves dissolve into spacetime.

Explain stress-energy tensor

General relativity energy"

I asked; What about covariance and energy?

"Covariance is the key reason why the old intuitive definitions of energy collapse in relativity — and why the relativistic definition is the only one that survives.What “covariance” demands-A physical law (or a physical quantity) must look the same in every inertial frame.
If two observers use different coordinates related by a Lorentz transformation, every equation must keep the same mathematical form (tensor equations). Energy in classical physics is a scalar that every observer agrees on (same number for total energy, same kinetic energy formula, etc.).
That cannot be true in relativity.Why classical energy is not covariant

• Classical total energy E_classical = ½mv² + … changes differently under Galilean boosts than under Lorentz boosts.
• Even the conserved quantity is different: under Galilean transformations the conserved energy transforms as
  E′ = E + mv·u + ½mu² (u = boost velocity)
  → different observers get different numerical values for “total energy” and even for what is conserved.

That is fine in Galilean relativity, but in special relativity the laws must be Lorentz-covariant, not Galilean-covariant.The only covariant way to define energyEnergy must be part of a 4-vector, because only then do the laws of physics keep the same form under Lorentz transformations.The energy-momentum 4-vector is
P^μ = (E/c, p_x, p_y, p_z)Under a Lorentz boost, E and p mix together exactly like t and x do in spacetime.
→ Different observers naturally measure different energies for the same system (just as they measure different times and lengths).Concrete consequences of covariance for energy

1. Energy is frame-dependent
   A photon has E = pc in every frame, but the actual value of E depends on the frame.
   A moving proton has much more energy than when it is at rest in your lab frame.
2. There is no universal “total energy of the universe”
   Because boosting the entire universe would change the total E, but not the physics.
   Only the invariant mass of an isolated system (√(E²/c⁴ − P_total²/c²)) is the same for everyone.
3. Conservation of energy is frame-dependent too
   Proper statement: In every inertial frame, the total 4-momentum is conserved.
   That automatically implies that the time component E/c is conserved in that frame, but the numerical value of the conserved energy is different in different frames.
4. You cannot separate kinetic, potential, and rest energy covariantly
   • Rest energy mc² → becomes part of total E in other frames.
   • Potential energy in electromagnetic or gravitational fields → mixes with kinetic and field energy under boosts.
     Only the total 4-momentum of particles + fields together transforms covariantly.

Bottom line-Covariance kills every classical intuitive notion of energy and forces the replacement:Classical picture → Relativistic (covariant) picture
Single scalar energy E → Component of the 4-vector P^μ
“Energy at rest = 0” → Invariant rest energy mc²
Kinetic/potential split → Not covariant, meaningless in general
Conservation of E → Conservation of entire P^μ in every frame. That is why textbooks say “energy and momentum are unified into a single 4-vector in relativity.”
It’s not just a mathematical trick — it’s the only way to have physical laws that are covariant under Lorentz transformations, i.e., the only way special relativity can be consistent."

A Theory Suggests the Universe Slowly Spins to Explain Hubble Tension

 It is philosophically interesting because of the problems of motion and the void. Modern physics and cosmology don’t acknowledge anything besides space-time and the physics of it. Yet in order to have the Universe rotating it need be in some volume or void with something beyond its edge. The intuitive idea of a primordial empty space holding material fields that appear to be space-time seems a pragmatic inference.

One body or universe cannot be said to ‘rotate’ unless in relation to something else. Even if it were in a true void, space-time would need to be said to rotate in relation to something more than nothing. If the two-dimensional fields emanating a third dimensions entanglement scenario were located on the event horizon of a spinning black hole, as some have suggested, perhaps the math could be worked out.

Obviously there may be some other explanation for the expansion of a two-dimensional ‘bubble’ of curvature within which a true void persisted even as a third dimension of entanglement occurred. Curvature with time added may appear to be rotation from some points of view. There is also the problem of the general relativity of a virtual, contingent three dimensional Universe of entanglement engaged via mass quantum entanglement and super-position naturally with a timeless two-dimensional substrate.

Interesting problems for philosophical thought- especially with the other new theory that time is a local phenomenon created by the relationship of quantum entanglement and super-position. Evidently the more quantum particles are entangled as at a black hole the more time is produced; no time in massless two-dimensional particles and vast amounts of time in black holes- unless black holes convert all of that quantum entanglement into passages to other parts of the Universe of space-time and white holes.

https://studyfinds.org/is-the-universe-spinning/

I looked into the matter. It’s not what is ‘in’ the Universe; rather it is what the Universe is contained in (such as an empty volume of nothing). It’s rotation could only be within itself and its own reference internally rather in relation to nothing outside. Apparently a Machian point of view with the rotation caused by internal forces and nothing else besides the Universe exist would be negated if the Universe were within a void of nothingness. In that case there might be no internal changes with the rotation 9at least if there was no central axis. If one modifies Newton’s spinning bucket experiment of a bucket spun at the end of a rope in absolute space of nothingness, and the bucket walls have no friction, this is what gemini said;

That’s an excellent modification to the thought experiment, highlighting the crucial role of friction (or viscosity in the case of a fluid).

If the bucket was absolutely without friction between its walls/base and the water:

Bucket Rotation: The rope spins the bucket, causing the bucket itself to rotate and move in a circular path through absolute space.

No Torque Transfer: Because there is absolutely no friction, the rotating walls of the bucket cannot exert any tangential force (torque) on the water. The walls simply slide past the water molecules they are in contact with.

Water’s Inertia (Rotational): The water, due to its inertia, will resist any change in its rotational state. Since no torque is applied to it by the frictionless bucket, the bulk of the water will not start rotating relative to absolute space (or its initial inertial frame). It remains rotationally static.

Water’s Motion (Translational): However, the water is contained within the bucket. As the bucket is forced by the rope to move in a circle, the physical boundary of the bucket will force the water as a whole to follow that same circular path through space. The bucket essentially pushes the water along this path. So, the water is accelerating centripetally (constantly changing direction). This centripetal force is provided by the pressure from the bucket walls pushing inward on the water mass.

Water Surface Shape: Since the water itself is not rotating (even though its center of mass is orbiting), there is no centrifugal effect within the water pushing it outwards towards the walls. The pressure distribution within the water remains uniform horizontally.—I asked Gemini about the Universe- it like your perspective… “If a universe was rotating in absolute space of nothingness, would anything internally change as a result of its rotation (there is no gravity or fields in nothingness of absolute space).

Show thinking

This is a fascinating question that delves into the very nature of space, motion, and inertia, touching on ideas from Newton, Mach, and Einstein. The answer depends significantly on which physical framework you adopt:

Newtonian View (Absolute Space):

Newton believed in “absolute space,” a fixed, immovable background stage against which all motion occurred. Rotation relative to this absolute space was “absolute rotation.”

In this view, if the entire universe (all its matter and energy) were rotating relative to this absolute space, then observers and objects within that universe would be in a non-inertial (rotating) frame of reference.

Yes, internal changes/effects would be observable.

Centrifugal Effects: Objects would experience an apparent outward force away from the axis of rotation. This could, for example, cause the large-scale structure of the universe to flatten into a disc over time, similar to how the spinning Earth bulges at the equator.

Coriolis Effects: Objects moving within the universe would be deflected from straight paths (as viewed from within the rotating frame). This affects weather patterns on Earth, for instance.

Sagnac Effect: Light travelling in opposite directions around a closed loop would take different amounts of time to complete the circuit, an effect measurable internally.

Essentially, according to Newton, the universe would “know” it’s rotating because physics behaves differently in a rotating frame compared to a non-rotating (inertial) one, irrespective of external reference points. Newton’s bucket experiment aimed to demonstrate this – the concave water indicates rotation relative to absolute space.

Machian View (Relationalism):

Ernst Mach disagreed with absolute space. He proposed (in what’s now called Mach’s Principle) that inertia itself (an object’s resistance to acceleration) arises from the object’s relationship or interaction with all other matter in the universe. Motion, including rotation, is only meaningful relative to other matter.

In this view, asking if the entire universe is rotating relative to absolute nothingness is potentially meaningless. If everything is rotating together, what is it rotating relative to?

Likely no, internal changes would not occur or the question is ill-posed. If inertia depends on the distribution of distant matter, and that entire distribution is rotating “as one,” there might be no internal reference frame against which to detect this rotation. The laws of physics inside would be the same as if it weren’t rotating.

Einstein’s General Relativity (GR):

GR replaces absolute space with dynamic spacetime, whose geometry is shaped by mass and energy. It incorporates some Machian ideas (like frame-dragging, where rotating masses drag spacetime) but doesn’t fully realize Mach’s Principle. Local inertial frames (where physics looks simplest, without fictitious forces) are determined by the local spacetime geometry, which is influenced by the overall distribution of mass-energy.

GR does allow for solutions representing rotating universes (like the Gödel Universe).

Yes, internal changes/effects would likely occur. In such models, the rotation affects the geometry of spacetime itself.

Observers would be able to perform local experiments (like using gyroscopes or observing light paths) to determine that they are in a globally rotating spacetime.

Frame-dragging would be a global effect.

The definition of a “non-rotating” frame locally would be tied to the rotating cosmic matter itself.

Conclusion:

Based on our current understanding of physics (leaning towards GR and the local validity of Newtonian concepts in weak fields/low speeds), if a universe containing matter and energy were rotating as a whole relative to some defined external standard (or if its own mass-energy distribution defined a state of overall rotation):

Yes, there would be internally observable consequences. Observers and experiments within that universe could detect this rotation through effects like centrifugal and Coriolis forces, the Sagnac effect, and potentially large-scale structural alignments or distortions caused by these effects over cosmic time. The universe wouldn’t need an external reference point to “know” it’s rotating; the physical laws operating within it would manifest differently compared to a non-rotating universe.” 

 I wonder if the entire Universe were rotating at the same speed, instead of around an internal axis of rotation, if that would make a difference. I think it interesting to consider the Universe as having an axis of rotation in space-time; the original spin before or after an inflaton, and how it would be effected by special relativity over time in regard to location in space-time history.

Idealism, Godel and Einstein

There is an epistemic and metaphysical difference of opinion about reality and the extent to which mind interprets and shapes its perception to conform to survival requisites. Instead of perceiving reality mind perceive a virtual box useful to it that is allowed by an unobserved reality. The topic is rather spacy and has numerous varieties.

Apparently Einstein too may have been something of an idealist for a while. Godel wrote about that in 1949. “Remark about the relationship between Relativity theorz and Idealistic Philosophy”.

Some believe that even gravity doesn’t exist in a heterodox relation with mind; that is, objectively?(is that even a valid term in the virtual box of perception paradigm)? They regard mind as shaping percepts that don’t actually reflect reality. Donald Hoffman is a recent developer of the theory.

If experience is a phenomenon of quantum Higgs field entanglements that ‘reality’ may be what is regardless of how it’s supported?

I need to read Godel’s paper on idealism and relativity. If God is hosting the show there is no way to discover how the metaphysics work.

To find out what the field configuration supporting the virtual box and gravity is will be very challenging for physicists. Perhaps researchers will go with what is possible starting from the evident.

I haven’t read Hoffman yet. I will get to that. In that paradigm one could theoretically turn off gravity if the perception/metavirtual box relation was understood.

https://www.udio.com/songs/ppYMShLZBG91focYi1NUDe

Maybe regard space time as a contiguous field that stretches like silly putty from a point to a vast pancake. The motion is space-time. Everything that happens in it are variegated particles of the field cycling through relationship protocols.

Lately there has been something of a return to idealism with mind interpreting reality in its own way for survival purposes. So in effect people experience apparent reality as if they were in a virtual box on a computer without actually experiencing the real reality.

That leaves a lot of room for fiction accounts of mechanics