Galadriel's Mirror uses transformation optics to mimic a curved spacetime that allows for closed timelike curves, enabling time travel. The presentation discusses: 1) Using a curved spacetime metric from general relativity that allows time travel, even if not physically realistic. Transformation optics can then create an equivalent material. 2) A curved spacetime example that tips light cones, making a path that circles the angular direction both null and closed, enabling light to travel in time. 3) The proposed mirror material would use this curved spacetime, curving light along a closed null curve that takes it into the past, allowing users to see into the future or past through the mirror.

1. Galadriel’s Mirror
Presentation by Reece Boston
April 23, 2014

2. Introduction to Galadriel’s Mirror
The proposed device is predicted to allow light to travel from the
future and into the past
What you will see, if you leave the Mirror free to work, I
cannot tell. For it shows things that were, and things that
are, and things that may yet be. But which it is that he sees,
even the wisest cannot always tell. Do you wish to look?
— Lady Galadriel, The Lord of the Rings
Begin with General Relativity, which allows for spacetimes that result
in time travel.
Transformation Optics: curved spacetimes −→ bi-anisotropic media
Bi-anistropy: D = E + γ1H, B = µH + γ2E.
We take a spacetime from GR that allows for time travel, and mimic it
using TO
Resultant material is Galadriel’s Mirror. Causes light to time travel.

3. Review of Transformation Optics in 3D
Maxwell’s Equations in a linear medium:
∂
∂xi
( ij
Ej ) = 0
∂
∂xi
(µij
Hj ) = 0
[ijk]
∂
∂xj
Ek = −
∂
∂t
(µij
Hj ) [ijk]
∂
∂xj
Hk =
1
c2
∂
∂t
( ij
Ej )
Maxwell’s Equations in curved space:
∂
∂xi
(
√
ggij
Ej ) = 0
∂
∂xi
(
√
ggij
Hj ) = 0
[ijk]
∂
∂xj
Ek = −
∂
∂t
(
√
ggij
Hj ) [ijk]
∂
∂xj
Hk =
1
c2
∂
∂t
(
√
ggij
Ej )
By comparing the two sets of equations, we find ij = µij =
√
ggij .
Light does not care. Curved space ⇔ linear medium.

4. Illustration: Cloak of Invisibility
In cylindrical coordinates, make transformation r → r = r−R1
R2
.
This will take origin, r = 0, and expand it into a cylinder of radius R1.
From Leonhardt’s paper

5. Illustration: Cloak of Invisibility
In cylindrical coordinates, make transformation r → r = r−R1
R2
.
This will take origin, r = 0, and expand it into a cylinder of radius R1.
The transformed space, r , θ , z , is a “curved” space, with spatial
metric tensor
gij =


a cos2 φ + b sin2
φ (a − b) cos φ sin φ 0
(a − b) cos φ sin φ a sin2
φ + b cos2 φ 0
0 0 1


for a = R2
R2−R1
3
and b = a r
r−R1
2
Light moving in the transformed space will not be able to reach the
area r < R1: that region is invisible.

6. More on Cloak of Invisibility
We take the curved space with metric tensor gij from previous slide.
We reinterpret the curved space, and insist that space is flat.
To make space flat and have same effect on light, we must also have a
linear material
ij
= µij
=
√
ggij
=


er cos2 φ + eφ sin2
φ (er − eφ) cos φ sin φ 0
(er − eφ) cos φ sin φ er sin2
φ + eφ cos2 φ 0
0 0 ez


for er = r−R1
r , eφ = r
r−R1
, and ez = R2
R2−R1
2
er .
Light will not care if we have medium , µ or curved space g
Both spaces will cause all objects in r < R1 to be invisible.

7. In summary...
We take a curved space describable by a metric tensor gij .
We may construct such a curved space through a transformation, but it
is gij that is important and not the transformation of space.
Using two sets of Maxwell Equations, we find a corresponding , µ.
Now we have a material that bends light in some desired way.
For the Cloak, the curvature is purely spatial; there is no need to
consider spacetime or magnetoelectric effect.
For the Mirror, exactly the same fundamental situation and
process...but in spacetime!
Invisibility cloak uses curved space with hidden region: the Mirror
uses curved spacetime with time-travel regions.

8. Time travel in General Relativity
This has long disturbed physicists by its refusal to be impossible.
Most succumb to “Garbage In/Garbage Out.”
The fundamental equation for General Relativity is the Einstein
Equation:
Rµν −
1
2
gµνR + Λgµν =
8πG
c4
Tµν
Relates the stress-energy density Tµν to the curvature of space in
Rα
µβν and gµν.
A particular gµν satisfying the above for a given Tµν is called a
solution.

9. Time travel in General Relativity
There exist solutions gµν and Tµν to the Einstein Equation that allow
for time-travel.
However: most of them require T that violate other rules of physics:
Wormholes require negative energy density
Tipler cylinders require infinitely-long cylinders
Godel space requires a slowly rotating universe, which is
empirically false
Kerr blackholes allow time travel, but only inside the event
horizon
The tools of GR very easily lend themselves to time-travel, so long as
we do not care about the physicality of Tµν or solving the Einstein
Equation...

10. ...And we don’t!
We are not General Relativists. We are Transformation Opticians!
Therefore we propose a physically absurd spacetime that will not
solve the Einstein Equation
. .. and will not arise from a physically realistic stress-energy T
. .. but it will allow for time travel. ...on paper and in our imagination.
Transformation Optics does not care about Tµν or the Einstein
Equation, only about gµν, therefore we can turn our imagination in to
reality.
Once I’ve shown that time-travel is possible with a given gµν, we use
TO to make it real in a material

11. Time-Like Curves
We start with the line element,
ds2
= dx2
+ dy2
+ dz2
− c2
dt2
which is a scalar invariant quantity. Also called metric function,
metric form.
The above equation gives infinitesimal spacetime distances along a
given worldline.
In the rest frame of any massive object, dx2 + dy2 + dz2 = 0, i.e. it is
not moving.
Therefore, for a massive object, in its rest-frame, ds2 = −c2dt2 < 0.
Such a spacetime distance is called “time-like.”
A “time-like curve” is a curve on which every spacetime distance is
time-like. It is also a curve that can be followed by a massive object.
Massive objects on time-like curves perceive themselves as moving
forward in time, i.e. experience positive proper time dτ = −ds/c.

12. Null or Light-Like Curves
For light, dx2 + dy2 + dz2 = c2dt2.
Therefore ds2 = dx2 + dy2 + dz2 − c2dt2 = 0.
Light experiences zero spacetime displacement and zero proper time.
Any displacement ds satisfying ds = 0 is called “null” or “light-like.”
A curve is null if all displacements along it are null displacements.
All massless objects must move on null curves.
Time-like curves and null curves are both classified together as
“causal curves,” because objects traveling on them can have causal
effect on events further down the world line.
The set of all null curves originating from an event in spacetime is
called the “light cone”, defines barrier of causal region

13. Space-like Curves
If two events have positive spacetime displacement, ds > 0, then they
are “space-like separated”.
This designation means that there does not exist any path in spacetime
that can bring a massive object from one event to the other.
Space-like separations are non-causal. Spatial distance is too great for
even light to traverse in given time.
If massive objets could be made to travel along space-like curves
(FTL), then time travel to the past would be very simple.
The difference between causal and non-causal is often illustrated with
the light cone.

14. The Light Cone and Causal Structure of Spacetime
(Presenter is not professional artist.)

15. The Light Cone and Causal Structure of Spacetime

16. The Light Cone and Causal Structure of Spacetime

17. The Light Cone and Causal Structure of Spacetime

18. Making the Unphysical Physical by Curving Spacetime
The pink trajectory on the last slide is unphysical in Minkowski space
Problem is orientation of light cones — up/down.
Think of spacetime curvature as tipping over light cones.
If we curve spacetime just right, we can make the pink trajectory to be
time-like by tipping over light cones so that the path is always inside a
light cone — i.e. is always time-like
If a time-like curve crosses itself, it is a Closed Time-like Curve
(CTC); such curves allow for time-travel in to the past.

19. Curved Spacetime Allowing Closed Time-Like Curves

20. Describing Curved Spacetime
It is useful to set c = 1.
Line element ds2 = −dt2 + dx2 + dy2 + dz2 = ηαβdxαdxβ.
ηαβ =




−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1



 −→ gαβ =




g00 g01 g02 g03
g10 g11 g12 g13
g20 g21 g22 g23
g30 g31 g32 g33




Line element ds2 = gαβdxαdxβ.
The g0i elements are what cause the light cones to tip along xi
direction.

21. Example of Tipped Light Cones: Alcubierre Warp-Drive
Metric function ds2 = −dt2 + (dx − v(r)dt)2 + dy2 + dz2
Almost flat. Function v(r) → 0 for r → ∞ and v(0) = vo.
Along r = 0, metric function ds2 = ds2
flat − 2vodxdt + v2
o dt2.
Last two terms are what tip light cone.
A massive object at r = 0 will have speed dx
dt = 1 + vo, which is
(globally) faster than light.
Locally, however, object moves with speed dx
dτ < 1 (i.e. is inside local
light cones).

22. Illustration of Alcubierre Warp-Drive
Light cones in warp-drive spacetime. Here D/T > 1 (Global FTL),
yet locally an observer that stays on thick black line (r=0) always
moves slower than light.

23. Closed Time-Like Curves
Here, as above, we tip the light cones, but along the angular direction.
We introduce metric form
ds2
= −dt2
+ dr2
+ dz2
+ 2f (r, φ)dφdt + r2
(1 − a/r)dφ2
.
Note simple case for dz = dt = dr = 0 and r < a: then
ds2
= r2
(1 − a/r)dφ2
< 0
This is a time-like curve: a massive object in this spacetime could
move along a curve of constant radius, height, and time.
Not useful for time travel though as curve does not traverse coordinate
time at all (dt = 0)

24. Illustration of Closed Time-Like Curves
(Presenter is not professional artist)

25. Making Closed Null Geodesics
In our case, we only care about behavior of light
Light moves on null geodesics, ds2 = 0
Note that if r = a and dr = dz = dt = 0, then
ds2
= r2
(1 − a/a)dφ2
= 0.
Not all null curves are null geodesics, seemingly in contrast to reason.
This point actually throws a slight wrench in the Mirror design...
A null geodesic satisfies
ds2
= 0 = gαβdxα
dxβ
(Null)
0 = d2
xα
+ Γα
βγdxβ
dxγ
(Geodesic)
We will need a special form of g0φ = gφ0 = f (r, φ)

26. ...a Fly in the Ointment
The Mirror was made under the assumption that all null curves are
geodesics.
Pretend this is the case, until recalculations are made.
The primary differences will be form of trajectory, t(φ) and the
metric element gφφ = r2(1 − a/r).
Besides not being a geodesic, everything else is correct in what
follows.
This is a closed null curve that light could follow in spacetime,
resulting in time travel. Just pretend that it is also a geodesic.

27. Using Closed Null Curves for Time Travel
Consider the curve r = a, z = 0, t(φ) = ∆t sin2 φ
2 + to. May as well
set to = 0.
Over 0 < φ < 2π, we now have 0 < t(φ) < ∆t, and
t(0) = t(2π) = 0.
For this to be null curve, 0 = −dt2 + 2f (r, φ)dφdt.
Function f (r, φ) unspecified; set to make equation work.
Rearranging
dt
dφ
= 2f (r, φ) = ∆t sin
φ
2
cos
φ
2
.
Therefore set f (r, φ) = A(r)∆t
4 sin φ, where A(r) is the radial part.
Now the curve t(φ) = ∆t sin2 φ
2 is a null curve by definition,
traversing total time ∆t

28. The Null Curve
t(φ) = ∆t sin2 φ
2

29. Let’s Stop Here
So Far:
How Transformation Optics works in 3D, taking curved space
gab and returning material ab, µab
Illustration of 3D Transformation Optics with Invisibility Cloak
Illustration of “tipped” light cones allowing for FTL in
Alcubierre warp-drive
Illustration of null and time-like curves in spacetime
ds2 = −dt2 + dr2 + dz2 + r2(1 − a/r)dφ2 + 2fdφdt
Next Time:
How Transformation Optics works in 3+1D, using Plebanski
equations
Derivation of ab, µab, γab
1 , γab
1 from gαβ
Discussion of physical feasibility of such a medium.
And eventually everything re-worked in terms of a Closed Null
Geodesic instead of a Closed Null Curve.