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."