Study-Unit Description

Study-Unit Description


CODE PHY3515

 
TITLE Gravitation and Cosmology

 
UM LEVEL 03 - Years 2, 3, 4 in Modular Undergraduate Course

 
MQF LEVEL 6

 
ECTS CREDITS 6

 
DEPARTMENT Physics

 
DESCRIPTION Gravity is the fundamental force that effects everything from dynamics on Earth to the evolution of the Universe. Newton's theory of gravitation explains to a very good extent the gravitational dynamics on Earth and within the solar system however it has several insurmountable inconsistencies and it does not agree with observations when scrutinized to sufficient accuracy. For this reason, Einstein formed his theory of General Relativity which is the current standard theory of gravity. In General Relativity the Newtonian concept of a gravitational field is replaced with the notion of the curvature of space-time. This can predict and explain phenomena such as the bending of light, perihelion precession of planetary orbits and gravitational red shift that have been known and well tested for a long time, as well as other phenomena such as gravitational waves which have recently been detected.

This study-unit will start with a review of Special Relativity, in which the concept of a four-vector and a tensor in flat space-time is developed. The Einstein's Equivalence Principle is introduced and used to explain the notion of curved space-time, which will lead to Einstein's field equations of General Relativity. The Shwarzschild black hole solution is described in detail. The rest of the study-unit looks at various applications of General Relativity in astrophysics, such as the classical tests, gravitational collapse and black holes, gravitational waves and their detection, and in cosmology with an emphasis on dark matter, dark energy and the early Universe.

Study-Unit Aims:

This study-unit will start with a review of Special Relativity, in which the concept of a four-vector and a tensor in flat space-time is developed. The Einstein's Equivalence Principle will be introduced and used to explain the notion of curved space-time. This is followed by the definition of the derivative on curved spacetime which will then be used to explain geodesics. Important tensors such as the Riemann curvature tensor, the Ricci tensor and the Einstein tensor, all of which are described in terms of the covariant derivative, will then be introduced. The Einstein's field equations of General Relativity are stated and compared to the equations of Newtonian gravity. The Schwarzschild black hole solution and its properties will be described in detail together with the associated classical tests of General Relativity.

The second part of the study-unit will centre on the application of general relativity to the domains of gravitational waves and cosmology. This will start by linearizing the gravitational field equations and exploring some wave-like solutions. Some properties of these waves will be investigated ranging from polarization to energy and wave detection. This section will then shift to cosmology where an introduction is given to the Friedmann–Lemaître–Robertson–Walker metric and the Friedmann equations together some selected solutions. The Hubble law and its use in modern cosmology will then be covered. In the context of this background, the very early Universe will then be connected to its currently accelerating state by an evolution between different epochs. This will introduce the ideas of cosmic microwave background radiation, large scale structure and the dark Universe.

Learning Outcomes:

1. Knowledge & Understanding:

By the end of the study-unit the student will be able to:

- apply Lorentz transformations to vectors and tensors;
- perform time dilation and length contraction calculations;
- use the conservation of four-momentum in relativistic particle mechanics;
- calculate the covariant derivative of vectors and tensors;
- obtain the geodesic equations in curved space-time;
- determine whether a space-time is flat or curved from the Riemann curvature tensor;
- obtain the Newtonian limit of Einstein's field equations;
- use the Schwarzschild metric to compute the angle of deflection of light, the angle of perihelion precession and the gravitational redshift;
- distinguish between the different polarization states of gravitational waves;
- apply the quadrupole formula to compute the energy loss due to gravitational radiation;
- understand the effect of gravitational waves on test particles and the design of gravitational wave interferometers;
- compute the evolution of the scale factor from the Friedmann equations for spatially flat or curved FLRW models;
- compute the luminosity distance, angular diameter distance and distance modulus and draw the Hubble diagram;
- compute the evolution of the density parameters;
- understand the need for an inflationary epoch in FLRW cosmology.

2. Skills:

By the end of the study-unit the student will be able to:

- apply Lorentz transformations to vectors and tensors between inertial frames;
- perform time dilation and length contraction calculations;
- write the Minkowski metric in different coordinate systems;
- use the conservation of four-momentum in relativistic particle mechanics;
- apply the Equivalence Principle to link flat and curved space-times;
- calculate the covariant derivative of vectors and tensors;
- obtain the geodesic equations in curved space-time;
- determine whether a space-time is flat or curved from the Riemann curvature tensor;
- quantify the amount of curvature in a given space-time;
- obtain the Newtonian limit of Einstein's field equations;
- use the Schwarzschild metric to compute the angle of deflection of light, the angle of perihelion precession and the gravitational redshift;
- distinguish between the different polarization states of gravitational waves;
- apply the quadrupole formula to compute the energy loss due to gravitational radiation;
- understand the effect of gravitational waves on test particles and the design of gravitational wave interferometers;
- compute the evolution of the scale factor from the Friedmann equations for spatially flat or curved FLRW models;
- compute the luminosity distance, angular diameter distance and distance modulus and draw the Hubble diagram;
- compute the evolution of the density parameters;
- understand the need for an inflationary epoch in FLRW cosmology.

Main Text/s and any supplementary readings:

Main Texts:

- Rindler W., 'Relativity - Special, General and Cosmological', Second Edition, Oxford University Press (2006)

Supplementary Readings:

- Schutz B.F., 'A first course in General Relativity' ,Second Edition, Cambridge University Press (2009)
- Cheng T.-P., 'Relativity, Gravitation and Cosmology, a basic introduction', Oxford University Press (2005)

 
ADDITIONAL NOTES Pre-requisite Qualifications: A good understanding of calculus and special relativity
Pre-requisite Study-units: PHY2210 and PHY2195 or PHY2295 and MAT2513

 
STUDY-UNIT TYPE Lecture and Tutorial

 
METHOD OF ASSESSMENT
Assessment Component/s Assessment Due Sept. Asst Session Weighting
Examination (3 Hours) SEM2 Yes 100%

 
LECTURER/S

 

 
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It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years.

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