Woei Chet Lim's
homepage
Office: G.3.09
Phone: (+64) 7 858 5148
Email: wclim@waikato.ac.nz
Teaching
MATH311-12A Advanced Calculus
MATH259-12B Mathematical Modelling
MATH542-12B Partial Differential Equations 2
MATH311-13A Advanced Calculus
MATH255-13B / ENGG284-13B Differential Equations
MATH333-13Y / MATH565-13B General Relativity
MATH311-14A Advanced Calculus
MATH255-14B / ENGG284-14B Differential Equations
MATH543-14B Nonlinear Dynamics and Chaos
(On study leave during the 15A semester)
MATH255-15B / ENGG284-15B Differential Equations
MATH333-15Y / MATH565-15B General Relativity
MATH541-15B Classical Partial Differential Equations
MATH311-16A Advanced Calculus
MATH541-16A Classical Partial Differential Equations
MATH255-16B / ENGG284-16B Differential Equations
MATH251-17A / ENGG285-17A Multivariable Calculus
MATH255-17B / ENGG284-17B Differential Equations
MATH333-17Y / MATH565-17B General Relativity
ENGG285-17T Multivariable Calculus
(On study leave during the 18A semester)
ENGEN201-18B Engineering Mathematics 2
MATHS201-18B Continuing Calculus (first 8 weeks)
MATHS203-18B Differential Equations and Modelling (last 4 weeks)
MATH541-18B Classical Partial Differential Equations
ENGEN201-19S Engineering Mathematics 2 (first half)
MATHS101-19A Introduction to Calculus
MATHS565-19A General Relativity
ENGEN201-19B Engineering Mathematics 2
MATHS201-19B Continuing Calculus (first 8 weeks)
MATHS203-19B Differential Equations and Modelling (last 4 weeks)
MATHS101-20A Introduction to Calculus
MATHS543-20A Nonlinear Dynamics and Chaos
ENGEN201-20B Engineering Mathematics 2
MATHS201-20B Continuing Calculus (first 8 weeks)
ENGEN201-20C (NET) Engineering Mathematics 2
ENGEN201-21A Engineering Mathematics and Modelling 2
MATHS201-21A Continuing Calculus (first 8 weeks)
MATHS565-21A General Relativity
(On study leave during the 21B trimester)
ENGEN201-22H (NET) Engineering Mathematics and Modelling 2
ENGEN201-22A Engineering Mathematics and Modelling 2
MATHS201-22A Continuing Calculus (first 8 weeks)
MATHS203-22B Differential Equations and Modelling
MATHS543-22B Nonlinear Dynamics and Chaos
ENGEN201-23H (NET) Engineering Mathematics and Modelling 2
ENGEN101-23A (HAM) Engineering Mathematics and Modelling 1A
ENGEN201-23A Engineering Mathematics and Modelling 2
MATHS201-23A Continuing Calculus (first 8 weeks)
MATHS565-23B General Relativity
ENGEN101-23B (HAM) Engineering Mathematics and Modelling 1A
(Tentative teaching allocation for 2024)
ENGEN201-24H (NET) Engineering Mathematics and Modelling 2
ENGEN101-24A (HAM) Engineering Mathematics and Modelling 1A / MATHS103-24A (HAM) Mathematics 1A
ENGEN201-24A Engineering Mathematics and Modelling 2 / MATHS201-24A Continuing Calculus
MATHS301-24A Real and Complex Analysis (complex half)
ENGEN101-24B (HAM) Engineering Mathematics and Modelling 1A / MATHS103-24B (HAM) Mathematics 1A
MATHS303-24B Applied Mathematics / MATHS503-24B Advanced Applied Mathematics
Research
My field of research is mathematical and computational cosmology.
My research involves modelling the universe as a whole, using a system of
partial differential equations (PDEs). As PDEs seldom admit exact
solutions,
I use dynamical systems and asymptotic analyses in simpler regimes of the
model, and perform numerical simulations in general regimes.
My research to-date has focussed on
the so-called inhomogeneous Mixmaster behaviour near a gravitational
singularity. A large part of my doctoral thesis is devoted to this topic.
The shock-capturing numerical code CLAWPACK, as well as two finite
difference codes developed by myself, were used to simulate inhomogeneous
Mixmaster behaviours, and results indicate that in a typical model,
the effect of inhomogeneity is negligible along most worldlines, except
for special worldlines where the effect becomes important from time to
time, and leads to recurring formation of spikes.
In 2007, I discovered the explicit OT G_2 spike solution.
Better numerical codes were then developed.
Numerical simulations of G_2 models confirm the recurrence of spikes over different Gowdy eras,
and that the explicit spike solution describes the spikes.
In 2015, I discovered the explicit non-OT G_2 spike solution, which generalizes the OT G_2 spike solution.
My other projects to-date include:
- Dynamics of spatially homogeneous (Bianchi) models
- Derivation and application of the boost formula
- Derivation of asymptotic expansions for cosmological models
- Analyzing singularities in exact solutions
My long-term goal is to investigate what the full nonlinear field
equations of general relativity predict about the evolution of
inhomogeneity.
Previous positions:
- Post-doctoral fellow (Geometric Analysis and Gravitation division)
Max Planck Institute for Gravitational Physics (Albert Einstein Institute). September 2008 -- December 2011.
- Post-doctoral fellow (with Frans Pretorius)
Department of Physics,
University of Alberta, Canada. September 2006 -- January 2007.
Department of Physics,
Princeton University, USA. February 2007 -- August 2008.
- Post-doctoral fellow (with Alan Coley)
Department of Mathematics and Statistics,
Dalhousie University, Canada. September 2004 -- August 2006.
Education:
- PhD Applied Math, University of Waterloo, Canada (2004)
Thesis:
"The dynamics of inhomogeneous cosmologies".
Supervisor: John Wainwright
- MMath Applied Math, University of Waterloo, Canada (2000)
Thesis: "Anisotropy of the cosmic microwave background
temperature in Bianchi universes".
Supervisor: John Wainwright
- BMath Honours Joint Applied Math & Computer Science,
University of Waterloo, Canada (1998)
Award:
-
S. Chandrasekhar Award
for the best post-doctoral presentation in all Mathematical General Relativity sessions
(A1 Exact Solutions and their Interpretation,
A2 Mathematical Relativity and Other Progress in Classical Gravity,
A3 Modified Gravity Theories) at the 19th International Conference on General Relativity
and Gravitation in Mexico City, July 5-9, 2010.
Link to ResearcherID page
Link to Google scholar page
Journal articles:
-
"Periodic boundary conditions and G_2 cosmology"
A.A. Coley, W.C. Lim
Classical and Quantum Gravity, 41, 015009 (12pp) (2024) ;
arXiv:2312.03285.
-
"Cosmological global dynamical systems analysis"
A. Alho, W.C. Lim, C. Uggla
Classical and Quantum Gravity, 39, 145010 (14pp) (2022) ;
arXiv:2201.07596.
-
"Numerical confirmations of joint spike transitions in G_2 cosmologies"
W.C. Lim
Classical and Quantum Gravity, 39, 065010 (16pp) (2022) ;
arXiv:2202.02480.
-
"Transition analysis of the non-OT G_2 stiff fluid spike solution"
W.C. Lim, M.Z.A. Moughal
Classical and Quantum Gravity, 39, 025010 (16pp) (2022) ;
arXiv:2111.15122.
Errata: n_3 in Eq (2) should be n_2. n_3^3 in Eq (4) should be n_3^2.
-
"Cylindrical spikes"
M.Z.A. Moughal, W.C. Lim
Classical and Quantum Gravity, 38, 075029 (22pp) (2021);
arXiv:2102.12028.
Note: This is an abridged version of Moughal's thesis, arXiv:2102.09776.
-
"Stiff fluid spike solutions from Bianchi type V seed solutions"
D. Gregoris, W.C. Lim, A.A. Coley
Classical and Quantum Gravity, 34, 235013 (21pp) (2017);
arXiv:1705.02747.
Errata: Reference [4] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"On the first G_1 stiff fluid spike solution in General Relativity"
A.A. Coley, D. Gregoris, W.C. Lim
Classical and Quantum Gravity, 33, 215010 (11pp) (2016);
arXiv:1606.07177.
Errata: Reference [17] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"Spikes and matter inhomogeneities in massless scalar field models"
A.A. Coley, W.C. Lim
Classical and Quantum Gravity, 33, 015009 (20pp) (2016);
arXiv:1511.07095.
Errata: In (39), the n_30 term should read n_30 exp(-(w+1)tau).
Reference [6] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"Non-orthogonally transitive G_2 spike solution"
W.C. Lim
Classical and Quantum Gravity, 32, 162001 (8pp) (2015);
arXiv:1507.02754.
Errata: In (10) and the line below it, n_10 and n_20 should be swapped.
In (16), the n_30 term should read n_30 exp(-(w+1)tau).
In (44), A^2 should be ln A^2.
Ignore equations (53)--(54). The last sentence in the paragraph before (49) should refer to n_10 and n_30.
Reference [15] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"Demonstration of the spike phenomenon using the LTB models"
A.A. Coley, W.C. Lim
Classical and Quantum Gravity, 31, 115012 (15pp) (2014);
arXiv:1405.5252.
Errata: Reference [4] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"General relativistic density perturbations"
W.C. Lim, A.A. Coley,
Classical and Quantum Gravity, 31, 015020 (13pp) (2014);
arXiv:1311.1857.
Errata: Reference [4] should be Lim et al 2009, PRD 79, 123526, not 103526.
-
"Spherically symmetric cosmological spacetimes with dust and radiation - numerical implementation"
W.C. Lim, M. Regis, C. Clarkson,
Journal of Cosmology and Astroparticle Physics, 10, 010 (17pp) (2013);
arXiv:1308.0902.
Errata: In (3.4) and (3.18), the 8/9 mu/G_+ v^2 term should have the opposite sign. Reference [15] should be Lim et al, PRD 79, 123526 (2009).
-
"The electromagnetic spike solutions"
E. Nungesser, W.C. Lim,
Classical and Quantum Gravity, 30, 235020 (11pp) (2013);
arXiv:1304.2964.
-
"Spike oscillations"
J.M. Heinzle, C. Uggla, W.C. Lim,
Physical Review D, 86, 104049 (26pp) (2012);
arXiv:1206.0932.
Errata: (2b) is incorrect. See Ref [51].
-
"Generating matter inhomogeneities in general relativity"
A.A. Coley, W.C. Lim,
Physical Review Letters, 108, 191101 (5pp) (2012);
longer version at arXiv:1205.2142.
-
"Future asymptotics of tilted Bianchi type II cosmologies"
S. Hervik, W.C. Lim, P. Sandin, C. Uggla,
Classical and Quantum Gravity, 27, 185006 (18pp) (2010);
arXiv:1004.3661.
-
"Properties of kinematic singularities"
A.A. Coley, S. Hervik, W.C. Lim, M.A.H. MacCallum,
Classical and Quantum Gravity, 26, 215008 (8pp) (2009);
arXiv:0907.1620.
-
"Spikes in the Mixmaster regime of G_2 cosmologies"
W.C. Lim, L. Andersson, D. Garfinkle, F. Pretorius,
Physical Review D, 79, 123526 (15pp) (2009);
arXiv:0904.1546.
Errata: In (A2), the exponent should have the opposite sign: -(tau-tau_0). In the last line of (B2), w should have absolute value sign.
-
"Evolution to a smooth universe in an ekpyrotic contracting phase with w >
1"
D. Garfinkle, W.C. Lim, F. Pretorius, P.J. Steinhardt,
Physical Review D, 78, 083537 (8pp) (2008);
arXiv:0808.0542.
-
"New explicit spike solution -- non-local component of the generalized
Mixmaster attractor"
W.C. Lim,
Classical and Quantum Gravity, 25, 045014 (17pp) (2008);
arXiv:0710.0628.
Errata: The third minus sign in (28) should be a plus. The factor 4
in (34) should not be there.
CQG 2007/2008 highlight paper.
-
"Late-time behaviour of the tilted Bianchi type VI-1/9
models"
S. Hervik, R.J. van den Hoogen, W.C. Lim, A.A. Coley,
Classical and Quantum Gravity, 25, 015002 (15pp) (2008);
arXiv:0706.3184.
-
"Late-time behaviour of the tilted Bianchi type VIh models"
S. Hervik, R.J. van den Hoogen, W.C. Lim, A.A. Coley,
Classical and Quantum Gravity, 24, 3859-3895 (2007);
gr-qc/0703038.
-
"Cosmic microwave background limits on spatially homogeneous
cosmological models with a cosmological constant",
A.A. Coley, W.C. Lim,
Classical and Quantum Gravity, 24, 889-895 (2007);
astro-ph/0610498.
-
"Kinematic and Weyl singularities",
W.C. Lim, A.A. Coley, S. Hervik,
Classical and Quantum Gravity, 24, 595-604 (2007);
gr-qc/0608134.
-
"Fluid observers and tilting cosmology",
A.A. Coley, S. Hervik, W.C. Lim,
Classical and Quantum Gravity, 23, 3573-3591 (2006),
gr-qc/0605128.
-
"Tilted Bianchi VII_0 cosmologies -- the radiation bifurcation",
W.C. Lim, R.J. Deeley, J. Wainwright,
Classical and Quantum Gravity, 23, 3215-3234 (2006);
gr-qc/0601040.
-
"Tilt and phantom cosmology",
A.A. Coley, S. Hervik, W.C. Lim,
Physics Letters B, 638, 310-313 (2006);
gr-qc/0512085.
-
"The late-time behaviour of vortic Bianchi type VIII Universes",
S. Hervik, W.C. Lim,
Classical and Quantum Gravity, 23, 3017-3035 (2006);
gr-qc/0512070.
CQG 2005/2006 highlight paper.
-
"Asymptotic silence-breaking singularities",
W.C. Lim, C. Uggla, J. Wainwright,
Classical and Quantum Gravity, 23, 2607-2630 (2006);
gr-qc/0511139.
-
"The futures of Bianchi type VII0 cosmologies with vorticity",
S. Hervik, R.J. van den Hoogen, W.C. Lim, A.A. Coley,
Classical and Quantum Gravity, 23, 845--866 (2006);
gr-qc/0509032.
-
"Asymptotic analysis of spatially inhomogeneous stiff and ultra-stiff
cosmologies",
A.A. Coley, W.C. Lim,
Classical and Quantum Gravity, 22, 3073--3082 (2005);
gr-qc/0506097.
-
"Asymptotic silence of generic cosmological singularities",
L. Andersson, H. van Elst, W.C. Lim, C. Uggla,
Physical Review Letters, 94, 051101 (4pp) (2005);
gr-qc/0402051;
Popular article "Quiet at the end"
by Davide Castelvecchi in Physical Review Focus,
15 , story 8 (2005).
-
"Isotropic singularity in inhomogeneous brane cosmological models",
A.A. Coley, Y. He, W.C. Lim,
Classical and Quantum Gravity, 21, 1311--1342 (2004);
gr-qc/0312075.
-
"Asymptotic isotropization in inhomogeneous cosmology",
W.C. Lim, H. van Elst, C. Uggla, J. Wainwright,
Physical Review D, 69, 103507 (22pp) (2004);
gr-qc/0306118.
-
"Anisotropic universes with isotropic cosmic microwave background
radiation",
W.C. Lim, U.S. Nilsson, J. Wainwright,
Classical and Quantum Gravity, 18, 5583--5590 (2001);
gr-qc/9912001.
-
"An almost isotropic cosmic microwave temperature does not imply an almost
isotropic universe",
U.S. Nilsson, C. Uggla, J. Wainwright, W.C. Lim,
The Astrophysical Journal, 522, L1--L3 (1999);
astro-ph/9904252.
Conference proceedings:
-
"General Relativistic Tilt and Dark Energy",
A.A. Coley, S. Hervik, W.C. Lim,
AIP Conf. Proc., 1083,
65--73 (2008).
-
"Cosmological models from a dynamical systems perspective",
J. Wainwright, W.C. Lim,
Journal of Hyperbolic Differential Equations, 2,
437--470 (2005);
gr-qc/0409082.
Essay competition:
-
"Cosmology: a bird's eye view", honorable mention in the Gravity Research
Foundation Essay Competition, 2006
A.A. Coley, S. Hervik, W.C. Lim,
International Journal of Modern Physics D, 15, 2187-2190
(2006),
gr-qc/0605089.
Conference and workshop presentations:
-
"Monotonic functions in Bianchi models with a scalar field",
presented at
- Joerg Frauendiener's 60th birthday conference, University of Otago, August 29, 2018.
- the NZMS Colloquium at University of Otago, December 4-6, 2018.
- the 10th Australasian Conference on General Relativity and Gravitation at Victoria University, Wellington, December 10-12, 2019.
-
"Spike dynamics near spacelike singularities",
presented at KTH, Stockholm, April 26, 2018, and at the 2nd workshop on "Singularities of general relativity and their quantum fate", Banach Mathematical Center, Warsaw, May 21-25, 2018.
-
"Exact G_1 spike solutions in general relativity",
presented at
- the 9th Australasian Conference on General Relativity and Gravitation at Gravity Discovery Centre, Gingin, November 27-30, 2017.
- the NZMS Colloquium at University of Auckland, December 5-7, 2017.
- the 6th ANZAMP meeting, Takapuna, January 30 - February 1, 2018.
-
"Spiky relativistic dynamics near spacelike singularities",
presented at the NZMS Colloquium at Victoria University, Wellington, December 5-7, 2016.
-
"Non-OT G_2 spike solution",
presented at
- the 8th Australasian Conference on General Relativity and Gravitation at Monash University, Melbourne, December 2-4, 2015.
- the 16th Canadian Conference on General Relativity and Relativistic Astrophysics in Vancouver, July 6-8, 2016.
- the 21st International Conference on General Relativity and Gravitation at Columbia University in New York City, July 10-15, 2016.
-
"Demonstration of the spike phenomenon using the LTB models",
presented at
- the Centennial GRG conference at Penn State University in State College, PA, June 8-12, 2015.
- the NZMS Colloquium at Massey University, Palmerston North, December 3-5, 2019.
-
"Spiky inhomogeneous exact solutions in general relativity",
presented at the Australian Institute of Physics Congress at ANU, Canberra, December 8, 2014.
-
"Tilt, spikes and generation of matter overdensities",
presented at the 7th Australasian Conference on General Relativity and Gravitation on Hamilton Island, December 10, 2013.
-
"BKL singularity, spikes and inhomogeneous cosmology",
presented at the MSRI programme "Mathematical General Relativity", November 26, 2013.
-
"Generating matter inhomogeneities in general relativity",
presented at the 13th Marcel Grossmann meeting in Stockholm, July 1-6, 2012.
Newer version presented at the 20th International Conference on General Relativity
and Gravitation in Warsaw, July 7-13, 2013.
-
"Dynamics near spacelike singularities is more inhomogeneous than conjectured by BKL",
presented at
- SFB seminar at AEI, October 25, 2010.
- Swedish GR meeting in Karlstad, December 3-4, 2010.
- University of Cape Town cosmology seminar, February 22, 2011.
- Workshop on "Dynamics of General Relativity: Analytical and Numerical Approaches",
at the Erwin Schroedinger Institute for Mathematical Physics, Vienna, July 20, 2011.
- The 6th Australasian Conference on General Relativity and Gravitation in Queenstown, February 11, 2012.
-
"Spike crossings in spacetimes with one Killing vector field",
presented at the 19th International Conference on General Relativity
and Gravitation in Mexico City, July 5-9, 2010.
Received the S. Chandrasekhar Award for the best post-doctoral presentation in all Mathematical General Relativity sessions
(A1 Exact Solutions and their Interpretation,
A2 Mathematical Relativity and Other Progress in Classical Gravity,
A3 Modified Gravity Theories).
-
"Spiky Mixmaster dynamics", presented at the 13th Canadian
Conference on General
Relativity and Relativistic Astrophysics in Calgary, May 16-18, 2009.
Also presented at the Atlantic Relativity Mini-Conference in
Fredericton, April 25, 2009;
at the 12th Marcel Grossmann meeting in Paris, July 13-18, 2009;
and at the "New Developments in Lorentzian Geometry" conference at TU Berlin, November 18-20, 2009.
-
"Spikes -- non-local component of the generalized Mixmaster attractor",
presented at the Geometric Analysis and Gravitation seminar at AEI, September 29, 2008.
Also presented at the workshop for Geometry, Analysis, and General Relativity,
at the Mittag-Leffler Institute, Stockholm, November, 2008.
Also presented at the Free University nonlinear analysis seminar, Berlin, July 15, 2010.
-
"Spikes in the Mixmaster regime of G_2 cosmologies",
presented at the 11th Eastern Gravity Meeting in State College, PA, May 12-13, 2008.
-
"Inhomogeneous Mixmaster and spikes", presented at the IGC inaugural
conference at Penn State University in State College, PA, August 9-11,
2007.
-
"Kinematic and Weyl singularities", presented at the 12th Canadian
Conference on General Relativity and Relativistic Astrophysics in
Fredericton, May 17-19, 2007.
Also presented at the 18th International Conference on General Relativity
and Gravitation in Sydney, Australia, July 8-13, 2007.
-
"Fluid observers and tilting cosmology", presented at the
Atlantic Relativity Mini-Conference in Fredericton, May 6, 2006.
Also presented at the 2nd International Meeting on Gravitation and
Cosmology in Santa Clara, Cuba, May 29-June 1, 2006, and at
the APCTP-TPI Meeting at the University of Alberta, December 18-23, 2006.
-
"Dynamical systems approach to inhomogeneous cosmology", presented at
the dynamical systems workshop for Global Problems in Mathematical
Relativity, September 12-25, 2005, at the Isaac Newton Institute for
Mathematical Sciences, University of Cambridge.
-
"Spiky spatial inhomogeneities", presented at the 11th Canadian
Conference on General
Relativity and Relativistic Astrophysics in Vancouver, May 19-21, 2005.
Also presented at the Atlantic Relativity Mini-Conference in Fredericton,
April 30, 2005.
-
"Asymptotic isotropization in inhomogeneous cosmology", presented at
the 10th Canadian Conference on General
Relativity and Relativistic Astrophysics in Guelph, May 28-31, 2003.
-
"Perturbations of the Friedmann-Lemaitre models: limitations and
pitfalls", presented at the 9th Canadian Conference on General
Relativity and Relativistic Astrophysics in Edmonton, May 24-26, 2001.
Maintained by Woei Chet Lim. Last updated 12 January, 2024.