• Skip to primary navigation
  • Skip to main content
SRI logo
  • About
    • Press room
    • Our history
  • Expertise
    • Advanced imaging systems
    • Artificial intelligence
    • Biomedical R&D services
    • Biomedical sciences
    • Computer vision
    • Cyber & formal methods
    • Education and learning
    • Innovation strategy and policy
    • National security
    • Ocean & space
    • Quantum
    • Robotics, sensors & devices
    • Speech & natural language
    • Video test & measurement
  • Ventures
  • NSIC
  • Careers
  • Contact
  • 日本支社
Search
Close
Publication January 1, 2010

Lorentzian Symmetry in an Expanding Hyperbolic Geometry: the Connection with Minkowski Space

Citation

Copy to clipboard


Smith, F. T. (2010). Lorentzian Symmetry in an Expanding Hyperbolic Geometry: The Connection with Minkowski Space. In Annales de la Fondation Louis de Broglie (Vol. 35, p. 23). Fondation Louis de Broglie.

Abstract

In a hyperbolic universe the 4-vectors and subspaces of velocity and position are isomorphic examples of Lorentzian symmetry and pseudospherical geometry. Light is shed on this connection by examining a hyperbolic 3-space as the subspace of an O(1,3) 4-space subject to the constraint of a pseudosphere of imaginary radius R = i! . This symmetry persists if the hyperradius of position space expands with cosmological time, as in the Hubble distance !H = ctH . Such an expanding non-euclidean space with a negative curvature changing in time leads automatically to the appearance of a differential 4-vector with an imaginary timelike component icΔt. This provides a new understanding of the origin of the time-varying coordinate in the Minkowski space-time four-vector. The position-velocity symmetry supports an enlarged Lorentz group based on the direct product of separate Lorentz subgroups in position and velocity spaces. The geometric isometry of negative curvature in both position and velocity spaces now provides additional evidence for a previously demonstrated reciprocal relationship between the Lorentz transformation phenomenology and the Hubble velocity-distance law.

↓ View online

Share this
Career call to action image

Work with us

Search jobs

How can we help?

Once you hit send…

We’ll match your inquiry to the person who can best help you.

Expect a response within 48 hours.

Our work

Case studies

Publications

Timeline of innovation

Areas of expertise

Institute

Leadership

Press room

Media inquiries

Compliance

Careers

Job listings

Contact

SRI Ventures

Our locations

Headquarters

333 Ravenswood Ave
Menlo Park, CA 94025 USA

+1 (650) 859-2000

Subscribe to our newsletter


日本支社
SRI International
  • Contact us
  • Privacy Policy
  • Cookies
  • DMCA
  • Copyright © 2023 SRI International
Manage Cookie Consent
To provide the best experiences, we use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us to process data such as browsing behavior or unique IDs on this site. Not consenting or withdrawing consent, may adversely affect certain features and functions.
Functional Always active
The technical storage or access is strictly necessary for the legitimate purpose of enabling the use of a specific service explicitly requested by the subscriber or user, or for the sole purpose of carrying out the transmission of a communication over an electronic communications network.
Preferences
The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user.
Statistics
The technical storage or access that is used exclusively for statistical purposes. The technical storage or access that is used exclusively for anonymous statistical purposes. Without a subpoena, voluntary compliance on the part of your Internet Service Provider, or additional records from a third party, information stored or retrieved for this purpose alone cannot usually be used to identify you.
Marketing
The technical storage or access is required to create user profiles to send advertising, or to track the user on a website or across several websites for similar marketing purposes.
Manage options Manage services Manage {vendor_count} vendors Read more about these purposes
View preferences
{title} {title} {title}