Where is the energy really? - ChatGPT

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Summary

So this puzzled me a lot when I studied physics at secondary school. When you throw a ball, it seems obvious that its energy travels with it. The teacher talks about its kinetic energy and you do the calculations. But modern physics gives a different picture. Energy is not an intrinsic substance stored inside objects. Reality is altogether more weird.

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The Energy Isn’t in the Ball

You see it soar through the air, a red-leathered projectile humming with menace. You raise your hands to catch it - thwack! - and for a moment the pain is tangible. So too, presumably, is the kinetic energy of the cricket ball. But where, precisely, was that energy located before it struck?

You might say, “It was in the ball, obviously.”

But was it?

In your frame - the one in which you were standing still and the ball was hurtling toward you - it certainly had kinetic energy, about half a joule for a fast delivery, enough to sting. Yet from the ball’s own point of view, it was sitting serenely at rest. It was your hands - indeed, the entire stadium and all the matter in the universe - that were flying toward it. In that frame, you had the energy.

Energy, as we shall see, is not a physical content carried by the ball but the value of the generator of time-translation symmetry in whatever frame we choose.

This is the first clue that something is amiss with our intuitive notion of “where” energy resides. In Newtonian mechanics we already learn that kinetic energy is frame-dependent. In special relativity, this becomes foundational: energy is just one component of the energy-momentum four-vector,

Pμ = (E/c, p_x, p_y, p_z).

This four-vector isn’t “inside” any object in an absolute sense; its components depend on your choice of inertial frame. In the ball’s rest frame, it is you who carries the kinetic energy. What remains invariant across all frames is the Minkowski norm,

E² − p²c² = m²c⁴.

No matter who computes it, the ball or the catcher, this quantity remains the same. For an isolated object considering itself stationary so that p=0, the invariant gives the rest mass. In your frame, you measure the ball’s energy and momentum, plug them into this equation, and again recover its mass. In the ball’s frame, energy is just mc² and momentum is zero—same result. The invariant has no spatial localisation; it is a scalar relation among dynamical quantities, not a distributable substance.

What slams into your hands is not a parcel of stored energy - it is a mismatch of frames, a disagreement about how to decompose four-vectors.

The cricket ball, calm in its own rest frame, surveys a universe of frenzied motion. You, by contrast, see the ball as the darting assassin. 

Yet both of you agree on the invariant: the square of the rest mass times c⁴. The physics holds; only the attributions vary. If you had been falling freely next to the ball, you’d have noticed no kinetic energy in it at all.

Energy and momentum are components of a single conserved object: the four-vector Pμ. This is constructed by integrating the stress-energy tensor T_μν over a spacelike hypersurface, your chosen slice of the universe at a constant time:

P^ν = ∫_Σ T^μν dΣ_μ.

When you, the catcher, compute the ball’s energy and momentum, you integrate over the spatial hypersurface t = 0 in your own frame. On this slice, the ball is moving and carries momentum p and corresponding kinetic energy.

In the cricket ball’s frame, however, things are reversed: it is at rest, and you are in motion. The ball’s computation of the total four-momentum of the rest of the universe - including your incoming body - is performed on a different slice of spacetime, the surface t′ = 0 in its own frame.

These slices are not the same. They carve up reality in distinct ways, intersecting worldlines differently.

Yet in both frames the conservation law

∂_μT^μν = 0

ensures consistency. Energy and momentum are not substances floating inside objects but properties of how stress-energy is distributed across your chosen spacelike slice. Change the slice, and you change the bookkeeping, though the physical predictions remain identical.

The invariant mass of each object is preserved even as energy and momentum vary with frame.

Under Lorentz transformations, the energy and momentum components of Pμ mix, just as time and space do:

E′ = γ(E − v p_x),    p′_x = γ(p_x − vE/c²).

A change of frame can convert what was previously momentum into energy and vice versa. The invariant norm remains constant, but its decomposition into energy and momentum depends on the observer. The total four-vector is conserved across all frames, yet the individual components - energy and momentum - are conserved only within a given inertial frame.

When you catch the ball, the energy it transfers to your hand is real enough, but it is not an absolute content bottled inside the leather. It is a frame-relative component of a conserved vector, evaluated on your slice of spacetime. From the ball’s perspective, you were the one carrying the energy.

So where does this four-vector itself come from? In field theory, it is an integral over a local density, the stress-energy tensor. The tensor T_μν represents the local density and flux of energy and momentum, derived from Noether’s theorem: invariance of the action under spacetime translations (x^μ → x^μ + a^μ) yields this conserved current.

For a general field theory with Lagrangian density ℒ(ϕ_i, ∂_μϕ_i),

T^μν = Σ_i (∂^μϕ_i) ( ∂ℒ / ∂(∂^νϕ_i) ) − η^μνℒ.

For a simple scalar field ϕ with ℒ = (1/2) ∂_μϕ ∂^μϕ − V(ϕ), this reduces to:

T^μν = ∂^μϕ ∂^νϕ − η^μνℒ.

The tensor is thus built from the fields and their derivatives, and the energy-momentum four-vector is simply its global integral. Beneath the tensor lies the Lagrangian density ℒ, which encodes the dynamics of the fields. It is not an observable; it is a compact rulebook, a generative algorithm constrained by symmetry and experiment.

The fields themselves - ϕ(x), A_μ(x), ψ(x), and, in general relativity, the metric g_μν(x) - appear to form the ontological ground of physics. Each of these fields assigns specific quantities to every point in spacetime:

• a scalar field ϕ(x) such as the Higgs field gives a single value;
• a vector field A_μ(x) describes directional quantities such as the electromagnetic potential;
• a spinor field ψ(x) represents matter with half-integer spin—electrons, quarks, neutrinos;
• the metric field g_μν(x) defines spacetime’s own geometry, setting distances, durations, and curvature.

Together, their interactions encode all known physical phenomena. They assign dynamical degrees of freedom to spacetime points. All observables and conserved quantities are constructed from them.

Yet modern theoretical work increasingly suggests that these fields, and spacetime itself, may be emergent from a deeper quantum-informational substrate. In holographic and tensor-network models, geometry arises from patterns of entanglement in Hilbert space. In such approaches, “fields in spacetime” are effective, macroscopic approximations of relations among underlying quantum degrees of freedom.

If this is right, the ontological descent does not end with fields. Beneath them lies the abstract structure of the Hilbert space and its symmetries. Energy then becomes the expectation value of the operator that generates unitary time evolution: the same conceptual role it played classically, now expressed within quantum dynamics.

Whether the relevant “time” is a coordinate in spacetime or an emergent internal parameter, the pattern is identical: symmetry first, conservation second, energy last.

Energy is not a substance to be found in the ball, nor anywhere else. It is the conserved Noether charge associated with the invariance of the laws under temporal translation.

What you feel when the ball strikes your palms is the local expression of a geometric symmetry of the universe - a momentary reconciliation of two incompatible foliations of spacetime. The sting is real, but the energy was never in the ball.

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