
Contents
-
-
-
-
-
-
-
-
-
-
-
11.1 Minkowski spacetime 11.1 Minkowski spacetime
-
11.1.1 Rotation in 3D space—a review 11.1.1 Rotation in 3D space—a review
-
11.1.2 The Lorentz transformation as a rotation in 4D spacetime 11.1.2 The Lorentz transformation as a rotation in 4D spacetime
-
-
11.2 Tensors in a flat spacetime 11.2 Tensors in a flat spacetime
-
11.2.1 Tensor contraction and the metric 11.2.1 Tensor contraction and the metric
-
The inverse basis vectors and the inverse metric The inverse basis vectors and the inverse metric
-
Inner products in terms of contravariant and covariant components Inner products in terms of contravariant and covariant components
-
-
11.2.2 Minkowski spacetime is pseudo-Euclidean 11.2.2 Minkowski spacetime is pseudo-Euclidean
-
Position and position derivatives Position and position derivatives
-
-
11.2.3 Relativistic velocity, momentum, and energy 11.2.3 Relativistic velocity, momentum, and energy
-
Four-velocity Four-velocity
-
Four-momentum Four-momentum
-
Massless particles always travel at speed c Massless particles always travel at speed c
-
-
11.2.4 The electromagnetic field tensor 11.2.4 The electromagnetic field tensor
-
11.2.5 The energy–momentum–stress tensor for a field system 11.2.5 The energy–momentum–stress tensor for a field system
-
The structure of the charge 4-current density The structure of the charge 4-current density
-
The structure of the energy–momentum–stress 4-current density The structure of the energy–momentum–stress 4-current density
-
Stress–energy tensor for an ideal fluid Stress–energy tensor for an ideal fluid
-
Nonrelativistic limit and the Euler equation Nonrelativistic limit and the Euler equation
-
-
-
-
11.3 The spacetime diagram 11.3 The spacetime diagram
-
11.3.1 Basic features and invariant regions 11.3.1 Basic features and invariant regions
-
11.3.2 Lorentz transformation in the spacetime diagram 11.3.2 Lorentz transformation in the spacetime diagram
-
Relativity of simultaneity, event-order, and causality Relativity of simultaneity, event-order, and causality
-
-
-
11.4 The geometric formulation—a summary 11.4 The geometric formulation—a summary
-
-
-
-
-
-
-
-
-
-
11 Geometric formulation of relativity
Get access-
Published:January 2013
Cite
Abstract
The relativistic spacetime invariant interval can be interpreted geometrically as the length in 4D Minkowski spacetime. The Lorentz transformation is a rotation in this pseudo-Euclidean manifold, with diag(-1,1,1,1) as the metric. This way the geometric formulation allows us to think of the metric as embodying all of special relativity. Tensors are quantities having definite transformation properties under a Lorentz transformation. If a physics equation can be written as a tensor equation, it automatically respects the relativity principle. Examples of tensors with one index (4-vectors) include the position 4-vector, with components of 3D positions and time, and momentum 4-vector with components of relativistic 3D momentum and energy. Examples of tensors with two indices (4-tensors of rank 2) include the energy–momentum–stress tensor (source term for the relativistic gravitational field) and the antisymmetric electromagnetic (EM) field tensor with the six components, being the (E,B) fields. The spacetime diagram is a particularly useful tool in understanding the causal structure the relativity theory.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMonth: | Total Views: |
---|---|
October 2022 | 1 |
November 2022 | 3 |
December 2022 | 2 |
April 2023 | 8 |
March 2024 | 2 |
April 2024 | 5 |
June 2024 | 1 |
September 2024 | 6 |
January 2025 | 2 |
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.