In this article, we argue that two distinct types of time should be taken
into account in relativistic physics: a geometric time, which emanates from the
structure of spacetime and its metrics, and a causal time, indicating the flow
from the past to the future. A particularity of causal times is that its values
have no intrinsic meaning, as their evolution alone is meaningful. In the
context of relativistic Lagrangian mechanics, causal times corresponds to
admissible parameterizations of paths, and we show that in order for a
langragian to not depend on any particular causal time (as its values have no
intrinsic meaning), it has to be homogeneous in its velocity argument. We
illustrate this property with the example of a free particle in a potential.
Then, using a geometric Lagrangian (i.e. a parameterization independent
Lagrangian which is also manifestly covariant), we introduce the notion of
ageodesicity of a path which measures to what extent a path is far from being a
geodesic, and show how the notion can be used in the twin paradox to
differentiate the paths followed by the two twins

Brunet, Olivier

English

arXiv

In this article, we argue that two distinct types of time should be taken into account in relativistic physics: a geometric time, which emanates from the structure of spacetime and its metrics, and a causal time, indicating the flow from the past to the future. A particularity of causal times is that its values have no intrinsic meaning, as their evolution alone is meaningful. In the context of relativistic Lagrangian mechanics, causal times corresponds to admissible parameterizations of paths, and we show that in order for a langragian to not depend on any particular causal time (as its values have no intrinsic meaning), it has to be homogeneous in its velocity argument. We illustrate this property with the example of a free particle in a potential. Then, using a geometric Lagrangian (i.e. a parameterization independent Lagrangian which is also manifestly covariant), we introduce the notion of ageodesicity of a path which measures to what extent a path is far from being a geodesic, and show how the notion can be used in the twin paradox to differentiate the paths followed by the two twins

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Geometric Time and Causal Time in Relativistic
Lagrangian Mechanics
Olivier Brunet
To cite this version:
Olivier Brunet. Geometric Time and Causal Time in Relativistic Lagrangian Mechanics. 2016. ￿hal-
01360468￿
Geometric Time and Causal Time in Relativistic
Lagrangian Mechanics
Olivier Brunet∗
September 5, 2016
In this article, we argue that two distinct types of time should be taken
into account in relativistic physics: a geometric time, which emanates from the
structure of spacetime and its metrics, and a causal time, indicating the flow
from the past to the future. A particularity of causal times is that its values
have no intrinsic meaning, as their evolution alone is meaningful. In the con-
text of relativistic Lagrangian mechanics, causal times corresponds to admissible
parameterizations of paths, and we show that in order for a langragian to not
depend on any particular causal time (as its values have no intrinsic meaning), it
has to be homogeneous in its velocity argument. We illustrate this property with
the example of a free particle in a potential. Then, using a geometric Lagrangian
(i.e. a parameterization independent Lagrangian which is also manifestly covari-
ant), we introduce the notion of ageodesicity of a path which measures to what
extent a path is far from being a geodesic, and show how the notion can be used
in the twin paradox to differentiate the paths followed by the two twins.
1 Introduction – The Two Relativistic Times
It is usually agreed that the theory of relativity has led one to merge space and
time into a single entity, spacetime, in which time and space should be considered
on par, the only difference between these two notions being the corresponding
sign in the signature of the metrics. But identifying time to a mere dimension of
spacetime is, in our opinion, hiding the fact that two types of time do actually
intervene and should be taken into account.
The first one is directly related to spacetime and to the corresponding di-
mension added to the three spatial ones. This kind ot time, which one usually
has in mind in a relativistic context, is essentially a “length” of a timelike in-
terval divided by the speed of light c. To that respect, it is directly related to
the geometry of spacetime through its metric, and we call it a geometric time.
Other typical examples of geometric times are those measured by the proper
time along a worldline or the time coordinate in a given Lorentz frame (again,
divided by c).
∗olivier.brunet at normalesup.org
1
However, another important type of time is present in relativity, related to
its causal structure and to the flow from the past to the future. This type of
time is a way to label the different stages during the temporal evolution of a
system. Usual examples of causal times are provided by the values of the indexes
labelling the hypersurfaces of a foliation of spacetime. Regarding worldlines, any
admissible parameterization provides a causal time, as long as tangent vectors
are future-oriented timelike ones. But contrary to geometric times, the values
of such a time have no intrinsic meaning, and it is the evolution of these values
alone which has a meaning as they help distinguish the future from the past.
Such times will be called causal.
Let us remark that in relativistic Lagrangian mechanics, both types of time
are present. First, as spacetime events are, in a given reference frame, de-
termined by for coordinates, one of which corresponding to a geometric time.
Thus, in the notation L(x, x˙), both x and the 4-vector x˙ have a component cor-
responding to a geometric time. However, the derivation leading to x˙ refers to
the dynamical evolution of x, so that it is made with regards to a causal time.
Similarly, in the Euler-Lagrange equation, the time derivation is also clearly
related to a causal time, rather than to a geometric one.
2 Parameterization Independance
Having identified two types of time in a relativistic setting, let us focus on
causal times and its role w.r.t. Lagrangian functions. As expressed previously,
the values of causal times do not have any intrinsic meaning. In particular, in
relativistic Lagrangian mechanics, as causal times correspond to parameteriza-
tions of paths, the fact that they should play no role in the physical behaviour
of a system implies that the action along a given path should not depend on the
actual admissible parameterization of the path.
In order to determine how this property translates in terms of Lagrangians,
consider a path P parameterized as x(t) for t ∈ [α, β]. The action of this path
is then
S(P) =
∫ β
α
L
(
x(t), x˙(t)
)
dt (1)
Any other parameterization of P can be written in the form y(u) = x ◦ ϕ(u)
with α = ϕ(a), β = ϕ(b), and such that ∀u ∈ [a, b], ϕ′(u) > 0. Considering this
second parameterization, we also want:
S(P) =
∫ b
a
L
(
y(u), y˙(u)
)
du (2)
This expression, in terms of x and ϕ, yields
S(P) =
∫ b
a
L
(
x ◦ ϕ(u), ϕ′(u) x˙ ◦ ϕ(u)) du (3)
2
But by an elementary change of integration variable (putting t = ϕ(u)) in
equation (1), we also obtain
S(P) =
∫ b
a
L
(
x ◦ ϕ(u), x˙ ◦ ϕ(u))ϕ′(u) du (4)
As we want both equations (3) and (4) to hold for any path and any suitable
function ϕ, this implies that L must be such that
∀λ > 0, L(x, λ x˙) = λL(x, x˙).
In other words, a Lagrangian has to be 1-homogeneous1 in its second argument
order to have causal times play no role in the value of the action along a path.
Obviously, as it is well known, this requirement has dramatic consequences
regarding Hamiltonian mechanics. We recall Euler’s homogeneous function the-
orem which states that a function is k-homogeneous if and only if, for all x,
x · ∇f(x) = kf(x)
If we define the conjugate momentum
pµ =
∂L
∂x˙µ
,
then Euler’s homogeneous function theorem entails pµx˙
µ = L, so that the Hamil-
tonian obtained as the Legendre Transform of L is
H = pµx˙
µ − L = 0
and, as Euler’s theorem states an equivalence, this means that a Lagrangian
function is parameter independent (i.e. causal time independent) iff the associ-
ated Hamiltonian is constantly zero. Conversely, having a non-zero Hamiltonian
means that the Lagrangian function relies on some particular causal time.
A consequence of this is that causal time independence implies that one
cannot rely on any Hamiltonian-based method, and should rely on Lagrangians
instead. And contrary, for instance, to Goldstein [GPS00] which states that
“there does not seem to be any compelling reason why the covariant Lagrangian
has to be homogeneous in the first degree”, we do believe that being causal time
independent is indeed a compelling reason, as for instance it ensures that the
action along a path depends only on the spacetime events constituting it, and
not on any particular parameterization of the path.
3 Geometric Lagrangians
Another natural requirement for a Lagrangian function in a relativistic setting,
aside being causal time independent, is to be manifestly covariant, so that it
1We recall that a function f is k-homogeneous if for all x and λ > 0, one has f(λx) =
λkf(x).
3
does not depend on a particular reference frame either. In that case, the value
of the Lagrangian would only depend on the geometry of a path, and not on
any particular curve having the path as its image and expressed in a particular
reference frame.
Definition 1 (Geometric Lagrangian) A Lagrangian will be said to be geo-
metric if it is both Lorentz-covariant, and 1-homogeneous in its second argument.
First, we can remark that the necessity of having a geometric Lagrangian
can be used as a guide for designing Lagrangians. In particular, any linear
combination of terms of the form
√
x˙µx˙µ, Aµ(x)x˙
µ,
1√
x˙µx˙µ
Bνη(x)x˙
ν x˙η,
1
x˙µx˙µ
Cνηκ(x)x˙
ν x˙ηx˙κ, etc.
leads to a geometric Lagrangian.
Consider, for instance, a free particle. It is possible to find a large variety
of Lagrangians for it in the literature. For instance, taken from [GPS00, JS98,
Rin06, HEL06], up to a multiplicative scalar constant, it is possible to find:
x˙µx˙
µ
√
x˙µx˙µ
√
1− x˙ix˙
i
c2
=
√
1− β2
where (x˙i), with i ranging from 1 to 3, represents the 3-speed of a particle in
the Lorentz frame “under consideration”. The presence of an additional (time
independent) potential energy usually leads to the addition of an extra term of
the form −U(x, y, z), leading to a Lagrangian like
−m
√
1− β2 − U(x, y, z)
However, quoting [JS98], “this treatment of the relativistic particle exhibits
some of the imperfections of the relativistic Lagrangian (and Hamiltonian) for-
mulation of classical dynamical systems. For one thing, it uses the nonrelativis-
tic three-vector velocity and position but uses the relativistic momentum. For
another, all of the equations are written in the special coordinate system in
which the potential is time independent, and this violates the relativistic prin-
ciple according to which space and time are to be treated on an equal footing.”
Considering the kinetic term alone, it is clear that
√
1− β2 is not covariant,
and that x˙µx˙
µ is not 1-homogeneous so that the only candidate for a geometric
Lagrangian is (with the correct multiplicative constants)
−mc
√
x˙µx˙µ
even though it might look “awkward” [Rin06].
For potential energy, a term of the form −U(x, y, z) is clearly not suitable
for a geometric Lagrangian. It is, in particular, not 1-homogeneous in x˙. But
it can easily be turned into a suitable geometric form the following way. Let eµ
be a vector basis for the Lorentz frame R in which V is defined. As it is an
4
energy, it can be seen as the time-component of a 4-vector. And, indeed, if one
defines:
A(ct, x, y, z) =
V (x, y, z)
c
e0,
it is then easy to verify that the term −Aµ(x) x˙µ leads to the correct equation
of motion using the geometric Lagrangian
−mc
√
x˙µx˙µ −Aµ(x) x˙µ = −mc
√
x˙ · x˙− V (x)
c
e0 · x˙ (5)
Geometric Lagrangians are also very closely related to the Euler-Lagrange
equation. Consider again the equality
L =
∂L
∂x˙
· x˙
verified by a geometric Lagrangian. If we differentiate this expression w.r.t.
parameter t, we get
∂L
∂x
· dx
dt
+
∂L
∂x˙
· dx˙
dt
=
( d
dt
∂L
∂x˙
)
· x˙+ ∂L
∂x˙
· dx˙
dt
⇐⇒ ∂L
∂x
· x˙+ ∂L
∂x˙
· x¨ =
( d
dt
∂L
∂x˙
)
· x˙+ ∂L
∂x˙
· x¨
⇐⇒ ∂L
∂x
· x˙−
( d
dt
∂L
∂x˙
)
· x˙ = 0
⇐⇒
(∂L
∂x
− d
dt
∂L
∂x˙
)
· x˙ = 0
In the final equality, we recognize
∂L
∂x
− d
dt
∂L
∂x˙
which appears in the Euler-Lagrange equation. As L is covariant, this expression
actually corresponds to a 4-vector, and the previous equality shows that it is
indeed orthogonal to x˙. In particular, it is spacelike and, as x˙ 6= 0, we have
∂L
∂x
− d
dt
∂L
∂x˙
= 0 ⇐⇒
∥∥∥∂L
∂x
− d
dt
∂L
∂x˙
∥∥∥ = 0
where we define the norm of a vector v as
∥∥v∥∥ = √∣∣vµvµ∣∣. This suggests the
following definition:
Definition 2 (Ageodesicity) The ageodesicity of a path P parameterized as{
x(t)
∣∣ t ∈ [a, b]} is the real number
Ag(P) =
∫
P
∥∥∥∂L
∂x
− d
dt
∂L
∂x˙
∥∥∥ dt
5
Let us first prove that the ageodesicity of a path is a purely geometric quan-
tity, i.e. it is covariant and does not depend on the parameterization of the path.
Indeed, with the previous notations, remembering that ϕ′(t) > 0 and since
∂2L is 0-homogeneous in its second argument, i.e. ∂2L(x, λx˙) = ∂2L(x, x˙)
2 (as
follows from the 1-homogeneity of L in its second argument), we have
∂1L
(
y(u), y˙(u)
)
= ∂1L
(
x ◦ ϕ(u), ϕ′(u) x˙ ◦ ϕ(u))
= ϕ′(u) ∂1L
(
x ◦ ϕ(u), x˙ ◦ ϕ(u))
and
[
u 7→ ∂2L
(
y(u), y˙(u)
)]′
(u) =
[
u 7→ ∂2L
(
x ◦ ϕ(u), ϕ′(u) x˙ ◦ ϕ(u))]′(u)
=
[
u 7→ ∂2L
(
x ◦ ϕ(u), x˙ ◦ ϕ(u))]′(u) = [(t 7→ ∂2L(x(t), x˙(t))) ◦ ϕ]′(u)
= ϕ′(u)
[(
t 7→ ∂2L(x(t), x˙(t))
)]′(
ϕ(u)
)
so that, considering the change of variable t = ϕ(u),
∥∥∥∂1L(y(u), y˙(u))− [u 7→ ∂2L(y(u), y˙(u))]′(u)
∥∥∥ du
=
∥∥∥ϕ′(u) ∂1L(x ◦ ϕ(u), x˙ ◦ ϕ(u))− ϕ′(u) [t 7→ ∂2L(x(t), x˙(t))]′(ϕ(u))
∥∥∥ du
=
∥∥∥∂1L(x ◦ ϕ(u), x˙ ◦ ϕ(u))− [t 7→ ∂2L(x(t), x˙(t))]′(ϕ(u))
∥∥∥ϕ′(u) du
=
∥∥∥∂1L(x(t), x˙(t))− [t 7→ ∂2L(x(t), x˙(t))]′(t)
∥∥∥ dt
With reasonable assumptions of continuity and of smoothness, a path has its
ageodesicity equal to 0 iff the vector ∂1L − ddt∂2L is null all along, i.e. iff it is
indeed a geodesic:
Theorem 1 A path P is a geodesic w.r.t. a geometric Lagrangian if and only
if it verifies
Ag(P) = 0
More generally, the ageodesicity of a path is a geometric measure of how far it
is from being a geodesic. In the next section, we will present an application of
this measure.
4 Ageodesicity for a Free Particle, and the Twin
Paradox
Let us consider again the geometric Lagrangian of a free particle (without po-
tential energy):
L = −mc
√
x˙ · x˙
2From here on, ∂1L corresponds to
∂L
∂x
, and ∂2L to
∂L
∂x˙
.
6
xt
1
v
c
λ→ 0+
λ
→
+
∞
Figure 1: Change of direction
In this situation, the Euler-Lagrange vector is simply
− d
dt
mcx˙√
x˙ · x˙
so that
Ag(P) =
∫
P
∥∥∥ d
dt
mcx˙√
x˙ · x˙
∥∥∥ dt
Let us compute the exact ageodesicity of a path corresponding to a change
of velocity. We consider the family of paths, as represented on figure 1, defined
for all t ∈ R⋆+ by
Xµ(t) = β
(
c(t− α
t
), vt, 0, 0
)
where α and β are parameters. In this case, one has
∥∥∥ d
dt
mcx˙√
x˙ · x˙
∥∥∥ = 2mvαtc2
(α+ t2)2c2 − v2t4 ,
one primitive of which being
mc
2
ln
(αc+ (c+ v)λ2
αc+ (c− v)λ2
)
Integrating from 0 to +∞, this leads to
Ag(P) = mc
2
ln
c+ v
c− v = mc arctanh
v
c
Up to the factor mc, we recognize the asymptotic change of rapidity. It can also
be remarked that this result neither depends on α nor on β, the two parameters
we had introduced for the sake of generality. And as, when β tends to 0, the
limit path corresponds to an instantaneous change of velocity of v as shown
7
β → 0
Figure 2: Change of direction, variation of β
in figure 2, this means that such a change of velocity does indeed lead to an
increase of ageodesicity of
mc arctanh
v
c
As an application of this result, let us consider the “twin paradox”. We
recall that this paradox involves two twins, one of whom makes a journey into
space in a high-speed rocket and returns home to find that his twin, who has
remained on Earth, has aged more. The puzzling aspect of this result is that
there seems to be a symmetry between the two twins, each of them seeing the
other as moving w.r.t. himself. However, this interpretation is rather naive, as
one twin remains in a single inertial frame while the trajectory of the other twin,
the one in the rocket, involves two frames: one for the outbound journey and
another for the inbound one.
In terms of ageodesicity, the two trajectories can easily be distinguished:
the twin who remains on Earth follows a geodesic (if we neglect the movements
she makes on Earth), so that the ageodesicity of its path is approximately 0.
Meanwhile, if the other twin starts its journey by travelling away from Earth
at speed v, and then travels back at the same speed, the change of direction
entails an ageodesicity of
mc arctanh
( 2vc
c2 + v2
)
(6)
The total ageodesicity associated to the travelling twin is even greater, as we
have neglected the ageodesicity contributions corresponding to the initial accel-
eration and the final deceleration.
This illustrates the fact that the ageodesicity of a path provides an intrinsic,
geometric measure of how far is a path from a geodesic and that, in the context
of the twin paradox, it provides a means to distinguish both trajectories.
5 Conclusing Remarks
In this article, we have just initiated the study of causal time in Lagrangian me-
chanics and introduced the notion of causal time independent Lagrangian and,
8
Ag = 0 Ag = mc arctanh
( 2vc
c2 + v2
)
Figure 3: Twin Paradox
more generally, of geometric Lagrangians. The latter notion, in particular, shall
prove to be especially useful in the context of relativistic Lagrangian mechanics.
The fact that any admissible parameterization shall lead to the correct result
(should it be an equation of motion, an action along a path or the ageodesicity
of a path) is a rather convenient property and the situation is, to that respect,
similar to the use of covariant expressions which ensures that the correct results
will be obtained, whichever reference obtained is used.
In the example of a free particle, it can be remarked that the parameteriza-
tion of the path was such that the parameter could hardly be seen as a geometric
time. However, it is an admissible causal time and thus, leads to the correct
actions and ageodesicities, as the Lagrangian is geometric.
In the case of multiple particles, the improvement of the situation is even
more evident. In order to have a covariant formulation, one usually imposes
the use of an affine parameter for a single particle, and this requirement cannot
be extended to a system of n particles, as it is usually not possible to define
a common affine parameter. The usual solution is, instead, to consider a pre-
ferred frame, at the cost of losing the generality of the result. But again, if the
considered Lagrangian function is geometric, one is assured to obtain general
and covariant results.
Finally, quoting Dirac [Dir33], “there are reasons for believing that the La-
grangian [formulation] is the more fundamental [than the Hamitonian one. In
particular,] the Lagrangian method can easily be expressed relativistically, on
account of the action function being a relativistic invariant; while the Hamilto-
nian method is essentially non-relativistic in form, since it marks out a particu-
lar time variable as the canonical conjugate of the Hamiltonian function.” We
do believe that one can go even further with this statement, as having a non-
vanishing Hamiltonien is precisely the sign that the corresponding Lagrangian
function is not causal time independent.
9
References
[Dir33] P. A. M. Dirac. The Lagrangian in Quantum Mechanics. Physikalische
Zeitschrift der Sowjetunion, 3, 1933.
[GPS00] Herbert Goldstein, Charles Poole, and John Safko. Classical Mechan-
ics. Addison-Wesley, 2000.
[HEL06] Michael P. Hobson, George P. Efstathiou, and Anthony N. Lasenby.
General Relativity, An Introduction for Physicists. Cambridge Univer-
sity Press, 2006.
[JS98] Jorge V. Jose´ and Eugene J. Saletan. Classical Dynamics: A Contem-
porary Approach. Cambridge University Press, 1998.
[Rin06] Wolfgang Rindler. Relativity. Oxford University Press, 2006.
10


Geometric Time and Causal Time in Relativistic Lagrangian Mechanics

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