What Is Energy & Is It Conserved?

Let’s take a tour from the very definition of energy in General Relativity, through the cheerful symmetries of anti-de Sitter space, and finally to the bleak reality of our dark-energy-dominated deSitter universe. Equations are your friends here - they make everything precise.

1. The notion of conserved energy in curved spacetime

1.1. Noether’s theorem meets geometry

In flat Minkowski space, energy is the conserved charge associated with time-translation invariance. The stress-energy tensor satisfies , and for any constant timelike vector the current

is conserved, . Integrating over a spacelike hypersurface gives a time-independent total energy.

In a general curved spacetime, the covariant conservation law still holds, but it does not imply a global conservation law unless there is a symmetry. To build a conserved charge you need a Killing vector field satisfying . Then

so the current is covariantly conserved and gives a conserved charge

where is a Cauchy surface with unit normal .

Thus global energy exists only when the spacetime admits a timelike Killing vector. Without one, the very concept of “total energy” becomes fuzzy.

1.2 Asymptotic rescue: ADM and Bondi Energy

Many physically relevant spacetimes (e.g. stars, black holes) are not globally stationary, but they become flat at spatial infinity. One can then define energy using the asymptotic time-translation Killing vector of Minkowski space. This leads to the ADM energy (Arnowitth-Deser-Misner):

where is the deviation of the spatial metric from flat space in asymptotically Cartesian coordinates. At null infinity, the Bondi energy gives a similar, but time-dependent, measure that accounts for gravitational radiation.

Both notions rely on the existence of an asymptotic timelike Killing vector. If space is not asymptotically flat (or asymptotically AdS), the construction fails.

2. Why energy is not conserved in de Sitter space

de Sitter space () is the maximally symmetric solution of Einstein’s equations with a positive cosmological constant . Its Ricci scalar is .

Global geometry: no global timelike Killing vector

The global metric of reads (setting the Hubble radius

This spacetime is a contracting-then-expanding 3-sphere. There is no globally timelike Killing vector: the only time-translation-like isometry is generated by , but it is timelike everywhere only in a small patch. In the global picture, becomes spacelike in the far past and future (the scale factor never turns around).

2.2 Static patch: a timelike Killing vector, but only a patch

One can write a static metric

which covers only the region (the static patch). Inside the cosmological horizon, is a timelike Killing vector, so one can define a conserved energy for observers who stay inside the static patch forever. However, this energy is not globally defined - it is a property of a particular family of observers, not of the whole spacetime. Moreover, any realistic cosmology involves degrees of freedom that eventually cross the horizon; from the global perspective, total energy is not conserved.

2.3 No Noether charge for the whole space

Because lacks a blogal timelike Killing vector, the covariant conservation does not integrate to a conserved charge over an entire Cauchy surface. The best one can do is use the conformal Killing vector of (e.g. the generator of dilations in the flat slicing) to build a “conformal energy”, but it is not constant in time.

2.4 Energy non-conservation in an expanding universe

The continuity equation for a perfect fluid in a FLRW universe,

shows that the energy density changes because of work done by the pressure. For vacuum energy () the equation is trivially satisfied, but any other matter component loses or gains energy from the gravitational field. There is no global energy to be conserved; the expansion of the universe breaks time-translation symmetry at a fundamental level. This is not a coordinate artefact - it is a genuine feature of spacetime with .

3. Why AdS/CFT works so beautifully (from the energy perspective)

Anti-de Sitter space (AdS) has a negative cosmological constant, . Its symmetry group is , which contains a global timelike isometry. This single fact makes AdS a paradise for defining conserved energy and for building a quantum theory of gravity.

3.1. Global AdS and its timelike Killing vector

The global metric of AdS is

where is the AdS radius. The Killing vector is timelike everywhere, so the spacetime is globally stationary. One can define a conserved energy

which is finite for suitable boundary conditions (after holographic renormalisation). This energy is the ADM energy of AdS, and it plays the role of the Hamiltonian in the bulk gravitational theory.

3.2. The boundary CFT and conformal generators

The conformal boundary of AdS is (for global coordinates). The isometry group of the bulk acts as the conformal group on the boundary. The bulk time translation maps exactly to the global time translation of the boundary CFT on the cylinder. Therefore the conserved energy in the bulk equals the conformal dimension (plus a possible Casimir energy) of the CFT state.

In radial quantisation of the CFT, the Hamiltonian on the sphere is dilatation operator , and the AdS energy is precisely the eigenvalue of . Thus the Hilbert and energy spectra match perfectly:

3.3 Why this is essential for AdS/CFT

A non-perturbative definition of quantum gravity requires a Hamiltonian that is bounded from below, a stable vacuum, and a well-defined partition function. AdS provides all three:

• Global timelike Killing vector -> conserved energy, ground state (empty AdS).
• Timelike conformal boundary -> a unitary CFT living on a compact space, with its own conserved energy.
• Hawking-Page phase transition -> the CFT can be heated to a deconfined plasma dual to a large AdS black hole, all controlled by the energy.

Without a conserved energy, the very idea of a dual Hamiltonian formulation would be ambiguous. The AdS/CFT dictionary maps the AdS energy to a CFT observable that is exactly conserved, giving the correspondence a solid quantum-mechanical footing.

4. But AdS is not our world - the hard truth about de Sitter

4.1. Observational reality

Cosmological data tell us that our universe is asymptotically de Sitter, not anti-de Sitter. The measured is positive (dark energy), so the far future is a dS space with Hubble constant . We do not live in a spacetime with a global timelike Killing vector, and energy is not globally conserved.

4.2. Why dS breaks the AdS/CFT template

• No global timelike Killing vector -> no conserved energy, no positive-definite Hamiltonian.
• Spacelike future infinity in dS, rather than a timelike boundary. This makes the natural “holographic screen” a spacelike surface, which is unsuitable for a standard unitary quantum mechanis.
• No stable ground state - dS space has a temperature (Gibbons-Hawking temperature , and there is no unique vacuum. The Hilbert space of a dS universe is notoriously hard to define.
• dS/CFT exists, but is conjectural and non-unitary: The dual theory would live on the future boundary, but because the boundary is spacelike, the “time” direction of the CFT must be taken as the renormalisation scale. The resulting CFT is Euclidean and typically contains operators with complex conformal dimensions. Unitarity is lost, and the holographic dictionary is far less rigorous than in AdS.

4.3. The tension with quantum gravity

String theory thrives in AdS because the boundary CFT provides a non-perturbative definition.For dS, we have no such formulation. Constructing a theory of quantum gravity in a universe with remains one of the greatest open problems. The absence of a conserved energy is a symptom of this deep difficulty.

Summary: a tale of three spacetimes

The beauty of AdS/CFT lies exactly in the fact that the bulk time-translation symmetry equips both sides of the duality with a conserved energy, making the quantum mechanical framework possible. The tragedy of our own universe is that its positive cosmological constant robs us of that symmetry, leaving us with a spacetime where energy is not conserved and holography is still an enigma.