Spacetime Geometry with Geometric Calculus

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
David Hestenes
Keyword(s):  
Geometric Calculus ◽  
Spacetime Geometry

Spacetime Geometry

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
Coordinate System ◽  
Special Relativity ◽  
Displacement Vector ◽  
Proper Time ◽  
Plane Geometry ◽  
Spatial Displacement ◽  
Spacetime Geometry ◽  
Space And Time ◽  
Displacement Vectors

This chapter shows that the counterintuitive aspects of special relativity are due to the geometry of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both x and y contribute positively to the distance, but in spacetime geometry the spatial displacement contributes negatively to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.

scholarly journals ON WEAK INTERACTIONS AS SHORT-DISTANCE MANIFESTATIONS OF GRAVITY

2013 ◽  
Vol 28 (07) ◽  
pp. 1350022 ◽  
Author(s):  
ROBERTO ONOFRIO
Keyword(s):  
Quantum Gravity ◽  
Short Distance ◽  
Weak Interactions ◽  
Planck Scale ◽  
Fundamental Symmetries ◽  
Spacetime Geometry ◽  
Newtonian Constant

We conjecture that weak interactions are peculiar manifestations of quantum gravity at the Fermi scale, and that the Fermi constant is related to the Newtonian constant of gravitation. In this framework one may understand the violations of fundamental symmetries by the weak interactions, in particular parity violations, as due to fluctuations of the spacetime geometry at a Planck scale coinciding with the Fermi scale. As a consequence, gravitational phenomena should play a more important role in the microworld, and experimental settings are suggested to test this hypothesis.

scholarly journals The art of the state

2018 ◽  
Vol 27 (11) ◽  
pp. 1843007 ◽  
Author(s):  
Christopher J. Fewster
Keyword(s):  
Infinite Family ◽  
Vacuum State ◽  
Natural Conditions ◽  
Spacetime Geometry ◽  
High Energies ◽  
Hyperbolic Spacetime ◽  
Hadamard Condition ◽  
Free Scalar ◽  
Model Independent ◽  
Hadamard States

Quantum field theory (QFT) on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed “SJ states”, that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that can yield an infinite family of Hadamard states, and also explain recent results that motivate the Hadamard condition without direct reference to ultra-high energies or ultra-short distance structure.

ΛCDM COSMOLOGY THROUGH THE LENS OF EINSTEIN'S STATIC UNIVERSE, THE MOTHER OF Λ

2013 ◽  
Vol 22 (03) ◽  
pp. 1350012 ◽  
Author(s):  
ABHAS MITRA ◽  
S. BHATTACHARYYA ◽  
NILAY BHATT
Keyword(s):  
Dark Energy ◽  
Milky Way ◽  
Fundamental Constant ◽  
Matter Density ◽  
Accelerated Expansion ◽  
Independent Manner ◽  
Static Universe ◽  
Spacetime Geometry ◽  
Λcdm Model ◽  
Do So

We show here that, in the context of Einstein's static universe (ESU), the static cosmological constant Λs = 0. We do so by extending (and not contradicting) the ESU relationship from Λs = 4πρ to Λs = 4πρ = 0, where ρ is the ESU matter density (G = c = 1). This extension follows from the fact that the elements of the spacetime geometry depend on pressure and energy density (ρ). Note in the ΛCDM model, Λ is associated with "Dark Energy (DE)." And, if Λ would be considered as a fundamental constant, it should be zero even for a dynamic universe. In such a case, the observed accelerated expansion could be an artifact of inhomogeneity [D. L. Wiltshire, Phys. Rev. D80 (2009) 123512; E. W. Kolb, Class. Quantum. Grav.28 (2011) 164009] or large peculiar acceleration of the Milky way [C. Tasgas, Phys. Rev. D84 (2011) 063503] or extinction of light of distant supernovae [R. E. Schild and M. Dekker, Astron. Nachr.327 (2006) 729, arXiv:astro-ph/0512236]. The same conclusion has also been obtained in an independent manner [A. Mitra, JCAP03 (2013) 007, doi: 10.1088/1475-7516/2013/03/007].

scholarly journals The confrontation between general relativity and experiment

2009 ◽  
Vol 5 (S261) ◽  
pp. 198-199
Author(s):  
Clifford M. Will
Keyword(s):  
General Relativity ◽  
Gravitational Wave ◽  
Equivalence Principle ◽  
Gamma Ray ◽  
Strong Field ◽  
Weak Field ◽  
X Ray ◽  
Spacetime Geometry ◽  
Tests Of General Relativity ◽  
Laser Interferometric

AbstractWe review the experimental evidence for Einstein's general relativity. A variety of high precision null experiments confirm the Einstein Equivalence Principle, which underlies the concept that gravitation is synonymous with spacetime geometry, and must be described by a metric theory. Solar system experiments that test the weak-field, post-Newtonian limit of metric theories strongly favor general relativity. Binary pulsars test gravitational-wave damping and aspects of strong-field general relativity. During the coming decades, tests of general relativity in new regimes may be possible. Laser interferometric gravitational-wave observatories on Earth and in space may provide new tests via precise measurements of the properties of gravitational waves. Future efforts using X-ray, infrared, gamma-ray and gravitational-wave astronomy may one day test general relativity in the strong-field regime near black holes and neutron stars.

GEOMETRY OF SPACETIME AND FINSLER GEOMETRY

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1678-1685 ◽  
Author(s):  
REZA TAVAKOL
Keyword(s):  
General Relativity ◽  
String Theory ◽  
Riemannian Geometry ◽  
Lorentz Invariance ◽  
Finsler Geometry ◽  
Spacetime Geometry ◽  
Test Particles ◽  
Light Rays ◽  
Recent Developments ◽  
Underlying Geometry

A central assumption in general relativity is that the underlying geometry of spacetime is pseudo-Riemannian. Given the recent attempts at generalizations of general relativity, motivated both by theoretical and observational considerations, an important question is whether the spacetime geometry can also be made more general and yet still remain compatible with observations? Here I briefly summarize some earlier results which demonstrate that there are special classes of Finsler geometry, which is a natural metrical generalization of the Riemannian geometry, that are strictly compatible with the observations regarding the motion of idealised test particles and light rays. I also briefly summarize some recent attempts at employing Finsler geometries motivated by more recent developments such as those in String theory, whereby Lorentz invariance is partially broken.

scholarly journals Quasar Clustering and Spacetime Geometry

1998 ◽  
Vol 498 (1) ◽  
pp. 11-25 ◽  
Author(s):  
Piotr A. Popowski ◽  
David H. Weinberg ◽  
Barbara S. Ryden ◽  
Patrick S. Osmer
Keyword(s):  
Spacetime Geometry
Sign in / Sign up

Export Citation Format

Share Document