Skip to main content

Department of Mathematics and Applied Mathematics

Department of Mathematics and Applied Mathematics

Search

  • Home
  • About Us
    • Vision
    • About
  • Undergraduate
    • Overview
    • First Year Course List
      • Overview and MAM Hotseat
      • Mathematics
      • Applied Mathematics
      • Engineering
      • Commerce
    • Second Year Course List
      • Overview
      • Mathematics
      • Applied Mathematics
      • Engineering
    • Third Year Course List
      • Overview
      • Mathematics
      • Applied Mathematics
      • Engineering
    • Forms & Documents
  • Postgraduate
    • Message from the Dean
    • Message from the Science Postgraduate Students' Association
    • Honours
    • Masters
      • Overview
      • AST5003F
    • PhD
    • Recent Graduates & Theses
    • Thesis Submission
    • Applications
      • Application to Tutor in the Department of Mathematics
    • Finances
    • Enquiries
    • Links
    • Contact Us
    • Forms & Documents
  • Staff
    • Forms & Documents
    • Permanent Academic Staff
    • Temporary Academic Staff
    • Administration
    • Emeritus Professors & Honorary Research Associates
  • Research
    • Research Areas
    • Centre for Research in Computational & Applied Mechanics
    • Cosmology & Gravity Group
    • Laboratory for Discrete Mathematics and Theoretical Computer Science
    • Laboratory for Quantum Gravity & Strings
    • Marine Resource Assessment & Management Group
    • Topology & Algebra Research Group
      • Introduction
      • Bernhard Banaschewski
      • Members
      • Activities
      • Output
      • Links
      • Conferences
      • Contact Us
    • National Astrophysics & Space Science Programme (NASSP)
    • Rangeland Modelling Group
      • Objectives
      • Future Projects
      • Publications
      • Contact Us
    • Recent Graduates & Theses
  • Outreach
    • Mathematical Digest
      • Overview
      • Archive
      • Articles
      • Problems
      • Humour
      • Links
      • Order
      • Contact
    • UCT Mathematics Competition
      • Overview
      • About the Competition
      • Maths Competition 2018
      • Challenge & UCT Mathematics Olympiad
      • Awards
      • Training Material
      • Entry Form
      • Registration & Invigilation
      • Winners
      • Photos
      • In the News
      • Credits
      • Contact Us
    • Mathematics Olympiads
      • International Mathematical Olympiad (IMO)
      • Pan African Mathematics Olympiad PAMO)
      • SA Mathematical Talent Search
    • Seminar Talks & Lectures
  • Employment
  • Contact Us
  • General Relativity Online Course
    • Overview
    • Tensors and Relativity: Chapter 1
      • Contents
      • Pre- relativistic physics
      • The equations of electromagnetism
      • The principle of Special Relativity
      • Spacetime diagrams and the Lorentz transformations
      • The spacetime interval
      • Minkowski spacetime
      • The null cone
      • Consequences of the Einstein postulates
      • Time dilation
      • Length contraction
      • The twin paradox
      • Velocity composition law
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 2
      • Contents
      • Four- vectors
      • Four- velocity, momentum and acceleration
      • Relativistic Doppler shift
      • Four- acceleration
      • Relativistic dynamics
      • Example 2.1
      • Example 2.2
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 3
      • Contents
      • Metrics and forms
      • One- forms
      • Gradients
      • The metric as a mapping of vectors onto one- forms
      • More general tensors
      • Tensors of type 0/2
      • Tensors of type 0/N
      • Tensors of type 2/0
      • Tensors of type M/0
      • Tensors of type M/N
      • Index ``raising'' and ``lowering''
      • General properties of tensors
      • Tensor derivatives and gradients
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 4
      • Contents
      • The energy- momentum tensor
      • General fluids
      • Conservation of energy- momentum
      • Conservation of particles
      • Perfect fluids
      • The conservation equations
      • The Electromagnetic tensor
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 5
      • Contents
      • The story so far
      • The gravitational redshift experiment
      • Non- existence of an inertial frame at rest on earth
      • Mass in Newtonian theory
      • The principle of equivalence
      • The principle of equivalence in action
      • Effect of gravity on light
      • Effect of gravity on time
      • Towards spacetime curvature
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 6
      • Contents
      • Manifolds, tangent spaces and local inertial frames
      • Covariant derivatives and Christoffel symbols
      • Calculating from the metric
      • Tensors in polar coordinates
      • Parallel transport and geodesics
      • The variational method for geodesics
      • The principle of equivalence again
      • The curvature tensor and geodesic deviation
      • The curvature tensor
      • Properties of the Riemann curvature tensor
      • Geodesic deviation
      • The Bianchi identities; Ricci and Einstein tensors
      • The Ricci tensor
      • The Einstein Tensor
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 7
      • Contents
      • Introduction
      • The non- vacuum field equations
      • The weak field approximation
      • Index
      • About this document ...
    • Tensors and Relativity: Chapter 8
      • Contents
      • The line element
      • The Christoffel symbols
      • The Ricci Tensor
      • The vacuum field equations
      • The Schwarzschild solution
      • General discussion of the Schwartzschild solution
      • Time dilation in a gravitational field
      • Length contraction in a gravitational field
      • Orbits in Schwartzschild spacetime
      • Solution for timelike orbits and precession
      • The bending of light
      • Index
      • About this document ...
    • Tensors and Relativity: Assignment 1
      • Problem
      • About this document ...
      • Hints 1
    • Tensors and Relativity: Assignment 2
      • Problem
      • About this document ...
      • Hints 2
    • Tensors and Relativity: Assignment 3
      • Problem
      • About this document ...
      • Hints 3
    • Tensors and Relativity: Assignment 4
      • Problem
      • About this document ...
      • Hints 4
    • Tensors and Relativity: Assignment 5
      • Problem
      • About this document ...
      • Hints 5
    • Tensors and Relativity: Assignment 6
      • Problem
      • About this document ...
      • Hints 6
    • Tensors and Relativity: Assignment 7
      • Problem
      • About this document ...
      • Hints 7
For information on South Africa's response to COVID-19 please visit the COVID-19 Corona Virus South African Resource Portal.
     Wolfram AlphaWebWork

Quick Links

Home > General Relativity Online Course > Tensors and Relativity: Chapter 4 > Conservation of particles
 
  • Overview
  • Tensors and Relativity: Chapter 1
  • Tensors and Relativity: Chapter 2
  • Tensors and Relativity: Chapter 3
  • Tensors and Relativity: Chapter 4
    • Contents
    • The energy- momentum tensor
    • General fluids
    • Conservation of energy- momentum
    • Conservation of particles
    • Perfect fluids
    • The conservation equations
    • The Electromagnetic tensor
    • Index
    • About this document ...
  • Tensors and Relativity: Chapter 5
  • Tensors and Relativity: Chapter 6
  • Tensors and Relativity: Chapter 7
  • Tensors and Relativity: Chapter 8
  • Tensors and Relativity: Assignment 1
  • Tensors and Relativity: Assignment 2
  • Tensors and Relativity: Assignment 3
  • Tensors and Relativity: Assignment 4
  • Tensors and Relativity: Assignment 5
  • Tensors and Relativity: Assignment 6
  • Tensors and Relativity: Assignment 7
Tensors and Relativity: Chapter 4

Conservation of particles

next up previous index
Next: Perfect fluids Up: The energy- momentum tensor Previous: Conservation of energy- momentum

It may also happen, that in a fluid flow, the total number of particles  does not change. This conservation law is derivable in the same way as the conservation of energy and momentum [ Assignment 4 ]:

equation824

or

equation826

We will confine ourselves to discussing fluids which obey this conservation law. This is hardly any restriction since we can always take n to be the number density of baryons.


Share on
 

Contact Details

Department of Mathematics and Applied Mathematics,
University of Cape Town,
Private Bag X1,
Rondebosch 7701,
South Africa.
Telephone: +27 (0)21-650-3191
Email: Hayley.Leslie@uct.ac.za

Quick Links

© University of Cape Town 2021. All rights reserved.