Eisenhart–Duval Lift: Geometric Embedding of Dynamics

• Eisenhart–Duval Lift is a geometrization method that embeds natural Hamiltonian dynamics into a higher-dimensional Lorentzian spacetime, replacing potentials with metric components.
• It transforms forced motion into free geodesic flow, enabling clear connections between classical mechanics, quantum reductions, and cosmological models.
• The framework unifies symmetry analysis and conformal geometry, revealing hidden conservation laws and offering new avenues for integrating dynamics with geometric structures.

Searching arXiv for recent and foundational papers on the Eisenhart–Duval lift. Searching arXiv for recent and foundational papers on the Eisenhart–Duval lift. The Eisenhart–Duval lift is a geometrization of classical dynamics in which a natural Hamiltonian system is embedded into a higher-dimensional Lorentzian spacetime so that the original trajectories arise as projections of null geodesics. In its standard Bargmann or Brinkmann form, the lift replaces scalar and vector potentials by metric components, introduces extra cyclic directions whose conserved momenta encode the lower-dimensional dynamics, and converts forced motion into free geodesic motion in an enlarged geometry. This construction has become a common framework for relating Hamiltonian mechanics, conformal geometry, hidden symmetries, Schrödinger and Dirac equations, minisuperspace quantum cosmology, and several cosmological model-building programs (Cariglia et al., 2015, Chanda et al., 2016, Dantas, 13 May 2026).

1. Geometric definition and canonical formulations

In the modern formulation, the lower-dimensional system is a natural Hamiltonian with quadratic kinetic term. For a scalar and vector potential one may write

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-eA_i\bigr)\bigl(p_j-eA_j\bigr)+e^2V(q),$$

and then homogenize it by adjoining two extra coordinates $u$ and $v$ with conjugate momenta $p_u$ and $p_v$. The lifted Hamiltonian becomes

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$

while the associated Lorentzian line element is

$$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$

Fixing $p_v=e$ and imposing the null condition $H=0$ yields $p_u=-\mathcal H/e$, and the physical time is recovered through $u$0 (Cariglia et al., 2015).

For a purely scalar potential and Euclidean configuration space, the same structure is often written in Brinkmann form as

$u$1

or equivalently

$u$2

The defining geometric feature is a parallel lightlike vector field, written as $u$3 or $u$4, which makes the lifted spacetime a pp-wave or Bargmann manifold (Aazami, 2024, Gibbons, 2020).

The literature also distinguishes a one-extra-coordinate Riemannian lift from the Lorentzian Eisenhart–Duval lift. The Riemannian version rewrites a potential as a positive metric component and therefore typically assumes $u$5, whereas the Lorentzian version is always well defined, naturally organized around null geodesics, and invariant under conformal rescalings of the metric along the null sector (Cariglia et al., 2015).

2. Projection to dynamics, time dependence, and the Jacobi–Eisenhart extension

The dynamical equivalence is encoded in the geodesic equations of the lifted metric. For the pp-wave metric

$u$6

the nonzero Christoffel symbols listed in the literature are

$u$7

Hence the geodesic equations take the form

$u$8

Choosing initial data so that $u$9 converts the last equation into

$v$0

which is exactly the original Newton or Hamilton equation $v$1 (Aazami, 2024).

This projection mechanism extends the older Jacobi–Maupertuis reinterpretation of fixed-energy trajectories as geodesics. For autonomous systems, the Jacobi metric

$v$2

reparameterizes the same trajectories on an energy shell. The obstruction for time-dependent systems is the absence of conserved energy. The Eisenhart–Duval lift resolves this by introducing an extra cyclic coordinate, denoted $v$3 in one formulation, with lifted line element

$v$4

Its conserved momentum

$v$5

replaces the missing conserved energy and permits a time-dependent Jacobi-type construction, referred to as the Jacobi–Eisenhart metric. In the non-relativistic limit this becomes

$v$6

so that non-autonomous dynamics again acquires a geodesic interpretation in an enlarged Lorentzian geometry (Chanda et al., 2016).

This suggests a general principle: the lift is not merely a change of variables, but a device for restoring geodesic structure precisely when the lower-dimensional system no longer carries the conserved quantities needed for a direct Jacobi–Maupertuis description.

3. Conformal geometry, symmetry algebras, and hidden structures

A central property of the Lorentzian lift is that null geodesics are preserved by conformal rescalings. In the simplest formulation this is expressed as

$v$7

which leaves the null trajectories unchanged. This is why conformal geometry, rather than only Riemannian geometry, is intrinsic to the Eisenhart–Duval framework (Cariglia et al., 2015).

One consequence is the possibility of classifying lower-dimensional systems by the conformal geometry of the ambient metric. For a one-dimensional system without vector potential, a general lifted metric can be written as

$v$8

Imposing flatness of the ambient metric forces the potential to satisfy

$v$9

so the admissible potential is at most quadratic,

$p_u$0

Under this condition the metric can be mapped to flat Bargmann or Minkowski form, thereby explaining both the free-particle/linear-potential correspondence and the harmonic-oscillator/free-particle correspondence as consequences of flatness of the Eisenhart metric rather than as isolated analytic transformations (Dhasmana et al., 2021).

The same geometric machinery organizes symmetry algebras. In gravitational minisuperspaces, the lifted metric

$p_u$1

allows conserved quantities to be obtained from conformal Killing vectors by $p_u$2. For flat FRW cosmology with a massless scalar and for Schwarzschild black-hole mechanics, the lifted metrics are conformally flat with 15 conformal Killing vectors forming $p_u$3, while the subset commuting with the null direction produces a centrally extended Schrödinger algebra $p_u$4 of observables (Achour et al., 2022).

The hidden-symmetry sector extends beyond ordinary Killing vectors. The lift of Killing–Yano and closed conformal Killing–Yano tensors provides symmetry operators for the Dirac equation, but the correspondence is selective: in the no-flux case there is a precise relation between higher- and lower-dimensional Dirac symmetries, whereas generic flux obstructs lifting or reducing all solutions and hidden symmetries (Cariglia, 2012). A further generalization replaces ordinary Lagrangian mechanics by Herglotz dynamics with an action-dependent Lagrangian. In that setting the generalized Brinkmann metric

$p_u$5

encodes the action variable $p_u$6, and infinitesimal symmetries of the reduced Herglotz system become conformal Killing vectors of the lifted metric (Bartczak et al., 29 Aug 2025).

4. Quantum reduction, Schrödinger equations, and Dirac-type formulations

The quantum counterpart of the lift begins from the same observation: the lower-dimensional potential becomes part of the ambient geometry. For a one-dimensional system, null reduction of the massless Klein–Gordon equation on the Eisenhart metric with ansatz

$p_u$7

gives the Schrödinger equation

$p_u$8

When the ambient metric is flat, the coordinate transformation that takes the Eisenhart metric to standard flat form also maps the wavefunction of the potential system to that of the free system,

$p_u$9

so the geometric flattening and the quantum phase factor are two aspects of the same construction (Dhasmana et al., 2021).

For spinors, dimensional reduction of the massless Dirac equation on the lifted Lorentzian manifold generally produces the non-relativistic Lévy–Leblond equation. In the special no-flux case $p_v$0 and $p_v$1, the reduction instead gives the lower-dimensional massive Dirac equation, yielding an exact reciprocal relation between lift and reduction. For generic scalar and vector fluxes, that reciprocity fails, and not all solutions or hidden symmetries survive the passage between dimensions (Cariglia, 2012). A related construction in Koopman–von Neumann mechanics uses an ultrahyperbolic Eisenhart–Duval metric on coordinates $p_v$2; null reduction of the massless Dirac equation then yields a Lévy–Leblond-type equation whose dynamical components satisfy the KvN evolution equation (Parida et al., 2023).

In minisuperspace quantum cosmology, the lift converts the Wheeler–DeWitt problem into a covariant equation on an extended configuration space. For FLRW cosmology with a homogeneous scalar field, one introduces an extra coordinate $p_v$3 and obtains the lifted metric

$p_v$4

so that the quantum constraint becomes

$p_v$5

In the same framework a first-order Dirac-type equation

$p_v$6

appears as the geometrically natural square root of the lifted Klein–Gordon or Wheeler–DeWitt operator (Kan et al., 2021).

A more explicit minisuperspace realization occurs in Rosen-Lagrangian cosmology. There the lifted two-dimensional configuration space with coordinates $p_v$7 carries metric

$p_v$8

and quantization leads to

$p_v$9

Because $$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$0 depends only on $$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$1, the wavefunction separates, and in the flat, cosmological-constant-dominated regime the scale-factor equation reduces to a Bessel equation with $$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$2, so the solutions become elementary trigonometric functions (Kaewkhao et al., 31 May 2026).

5. Cosmology and minisuperspace model building

The lift has been used extensively in cosmology because minisuperspace systems are naturally finite-dimensional and often constrained. For a minimally coupled scalar field in flat FLRW spacetime, one Riemannian formulation introduces an auxiliary coordinate $$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$3 and the lifted metric

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$4

In three dimensions, conformal flatness is equivalent to vanishing Cotton–York tensor, and that condition singles out

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$5

as the unique potential for which the Eisenhart metric is conformally flat. In adapted variables and with a suitable lapse choice, the geodesic equations then linearize to

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$6

while the lifted Wheeler–DeWitt equation becomes the flat-space wave equation

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$7

The enlarged quantum system has additional solutions unless a reduction condition is imposed to recover the original cosmological sector (Paliathanasis, 2024).

In scalar–tensor gravity, the same strategy becomes an integrability criterion. The lifted Lagrangian with cyclic variable $$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$8 produces the extended minisuperspace metric

$$H=\frac12 \sum_{i,j=1}^n h^{ij}(q)\bigl(p_i-p_vA_i\bigr)\bigl(p_j-p_vA_j\bigr)+p_v^2V(q)+p_up_v,$$9

Demanding conformal flatness, again through the Cotton–York tensor, yields two principal admissible families: $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$0 or

$$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$1

In the corresponding variables the geodesic equations reduce to the linear system $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$2, which then generates explicit cosmological solutions in Brans–Dicke theory, $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$3 gravity, and hybrid metric–Palatini models (Paliathanasis, 16 Jan 2025).

Scalar-field cosmology with dust provides another version of the same program. There the lifted Hamiltonian includes auxiliary momenta $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$4, $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$5, and $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$6, and the classification of lifted conformal symmetries selects the potential

$$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$7

The lifted Killing generators descend to nonlocal conservation laws in the original cosmology, while a separate Painlevé analysis shows that the quintessence system has the Painlevé property only for $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$8, whereas the phantom model is integrable for any $$ds^2=\sum_{i,j=1}^n h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr).$$9 (Paliathanasis, 26 Jun 2025).

The cosmological Eisenhart–Duval program is not restricted to minisuperspace quantization. A genuinely cosmological Bargmann metric,

$p_v=e$0

builds the cosmic scale factor directly into the lifted geometry. With a null-fluid stress tensor, Einstein’s equations reduce to the Ermakov–Milne–Pinney equation

$p_v=e$1

and the same setting yields Newton–Hooke boosts, null translations, the Ermakov–Lewis invariant, and a lifted description of the Dmitriev–Zel'dovich equations for particles in an expanding universe (Cariglia et al., 2018).

Rosen’s Newtonian cosmology furnishes a further application in which the cosmological constant is promoted to

$p_v=e$2

The lift introduces an auxiliary coordinate $p_v=e$3, the conformal Killing equations yield

$p_v=e$4

and the Wheeler–DeWitt equation becomes analytically solvable in the cosmological-constant-dominated regime. The construction recovers $p_v=e$5CDM at $p_v=e$6, while the physically viable quantum sector identified in that analysis is $p_v=e$7 (Kaewkhao et al., 31 May 2026).

6. Generalizations, embeddings, and structural limits

The lift extends beyond point mechanics. For homogeneous field theories with fields $p_v=e$8, one adds a fictitious scalar $p_v=e$9 and writes

$H=0$0

The potential is absorbed into the extended field-space metric, and Noether symmetries of the original theory become Killing vectors of the lifted field space. For spacetime-dependent field theories, a single fictitious scalar is no longer sufficient; the consistent extension uses a fictitious vector field $H=0$1 and a mixed vielbein, producing a kinetic-only theory whose geometry is not a simple sigma model (Finn et al., 2018).

The lifted geometry itself may admit a higher-dimensional interpretation. For Euclidean configuration space and in the absence of magnetic-type forces, the Eisenhart–Duval spacetime can be embedded as an $H=0$2-brane in a flat $H=0$3-dimensional ambient space with signature $H=0$4. If the lifted metric is Ricci-flat, equivalently if $H=0$5, the embedded brane extremizes its spacetime volume. Newtonian $H=0$6-body gravity provides the concrete example of a $H=0$7-brane moving in flat $H=0$8-dimensional space with two times (Gibbons, 2020).

The notion of lift can also be generalized by promoting coupling constants to extra-dimensional momenta. In the Toda chain this yields a multi-coordinate extension in which each $H=0$9 becomes a conserved momentum $p_u=-\mathcal H/e$0, so the generalized Hamiltonian

$p_u=-\mathcal H/e$1

acts as an inverse Kaluza–Klein reduction. The standard Eisenhart metric is then recovered as a dimensional reduction of the generalized lift, and the lifted Lax invariants become higher-rank Killing tensors (Cariglia et al., 2013).

Higher-derivative theories admit only qualified analogues. A direct Ostrogradsky-based generalization introduces additional auxiliary sectors and can reproduce the target dynamics only for restricted classes of homogeneous potentials or after canonical transformations that remove the momentum-linear obstruction. In those constructions the resulting metrics are generally ultrahyperbolic rather than Lorentzian, and they do not retain all the special properties of the standard Brinkmann lift (Galajinsky et al., 2016).

The principal structural limitation is that the standard Eisenhart–Duval lift requires a natural Hamiltonian,

$p_u=-\mathcal H/e$2

because the geodesic Hamiltonian of any finite-dimensional metric is necessarily quadratic,

$p_u=-\mathcal H/e$3

Consequently, single-degree-of-freedom minisuperspace Hamiltonians that are not quadratic in canonical momenta do not admit an Eisenhart–Duval lift. Effective Loop Quantum Cosmology models with polymer-modified Hamiltonians of the form

$p_u=-\mathcal H/e$4

therefore lie outside the metric ED framework. The literature identifies this as a kinematical no-go theorem and points to possible extensions beyond the ordinary metric category, including Finsler geometry, although no canonical Finsler analogue of the lift is known (Dantas, 13 May 2026).