Deriving Einstein's Field Equations of General Relativity PDF

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General Relativity and Solutions to Einstein’s Field Equations Abhishek Kumar Department of Physics and Astronomy, Bates College, Lewiston, ME 04240 General Relativity and Solutions to Einstein’s Field Equations A Senior Thesis Presented to the Department of Physics and Astronomy Bates College in pa...

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General Relativity and Solutions to Einstein’s Field Equations

Abhishek Kumar Department of Physics and Astronomy, Bates College, Lewiston, ME 04240

General Relativity and Solutions to Einstein’s Field Equations A Senior Thesis Presented to the Department of Physics and Astronomy Bates College in partial fulfillment of the requirements for the Degree of Bachelor of Arts by

Abhishek Kumar Lewiston, Maine January 13, 2009

Contents Acknowledgments

iii

Introduction

iv

Chapter 1. Einstein’s Concept of Gravity

1

Chapter 2. Consequences of General Relativity

3

1. Gravitional Lensing

3

2. Black Holes

4

3. The Expanding Universe

7

Chapter 3. Mathematical Tools of General Relativity

10

1. Indices and Summation Convention

10

2. Defining Metrics and Tensors

10

3. Vectors in Curved Spacetime and Gradients

11

Chapter 4. The Einstein Curvature Tensor

14

1. Introduction to Curvature Tensors

14

2. The Schwarzschild Metric

15

3. The Robertson-Walker Metric

22

Bibliography

30

ii

Acknowledgments First and foremost, I would like to thank my wonderful thesis advisor, Professor Tom Giblin. The high level of technicality required to understand a subject like General Relativity requires quite an amount of patience. Professor Giblin has been exceedingly understanding and supportive in helping me understand Einstein’s beautiful theory. I would also like to acknowledge my father, mother and brother, who were supportive throughout my efforts. My family helped mitigate the frustration that sometimes arose with not obtaining the correct results or if calculations were not going my way. And when results were coming out favorably, my family was also there to congratulate me. The Physics Department also deserves an overwhelming thank you for putting together such a wonderful and understanding team of professors. My skills in Physics have grown to a point where I can begin comprehending General Relativity only because of the excellent teaching of the professors in the Physics Department.

iii

Introduction Albert Einstein’s theory of General Relativity has revolutionized modern physics. General Relativity is a theory that relates space and time to gravitation. Einstein’s theory of General Relativity challenged Isaac Newton’s theory of gravity. In Newtonian gravity, space and time were absolute; they are unchangeable and completely static. In Einsteinian gravity, however, space and time are intertwined into one single fabric, that can be distorted and manipulated. This distortion is what creates the phenomenon known as gravity. With the introduction of Einstein’s theory of General Relativity, various new doors began opening up in physics. We were able to better understand various phenomenon occuring as a consequence of General Relativity. Some examples are gravitational lensing, black holes and the expansion of our universe. The explanation of these phenomenon can be found within Einstein’s Field Equations. This paper aims to show how the Einstein’s gravity works differently from Newtonian gravity, while providing substantial discussion on various consequences of his theory. The paper then seeks to provide the reader with various mathematical tools required to understand Einstein’s Field Equations. Lastly, the paper will show the calculation of Einstein’s curvature tensor for metrics modeling phenomenon such as non-rotating black holes and the expanding universe.

iv

CHAPTER 1

Einstein’s Concept of Gravity The concept of gravity was truly first understood by Isaac Newton. The famous incident of him sitting under a tree and an apple falling down in front of him led him to unite the motion of terrestrial objects and celestial objects in a single theory he called Gravity. He postulated that the force that keeps us on the ground, made that apple fall to the ground and the force that keeps the planets in orbit around the sun are both the same. In other words, any two masses essentially attract each other with a force that is given by the formula, (1.1)

Fgravity =

Gm1 m2 , 2 r12

2 where m1 and m2 are the masses of the first and second mass, respectively, r12 is the

distance between the two masses, and G is Newton’s Gravitational Constant. This theory remained the prominent theory of gravity for more than two-hundred years until Albert Einstein proposed his theory of gravity. Einstein’s rumminations on light while forming his theory of Special Relativity led him to find a dire contradiction in Newton’s theory of gravity. Einstein discovered that nothing can travel faster than the speed of light; the speed of light is essentially a universal speed limit. However, Newton’s gravity given in Equation (1.1) has no timedependence. The gravitational force of one mass on another acts instantaneously. Say, for instance, if the sun in our solar system were to, hypothetically, vanish, Newton’s time-independent version of gravity says that the planets would immediately spin out of orbit. This, however, cannot be true Einstein thought because if nothing can travel faster than the speed of light, how can a gravitational affect travel faster than 1

1. EINSTEIN’S CONCEPT OF GRAVITY

2

the speed of light? To solve this issue, Einstein developed the General Theory of Relativity, where gravity was not instantaneous, but its effects traveled at the speed of light. Einstein proposed a revolutionary idea in which the three dimensions of space and time were intertwined in one single four-dimensional fabric that he would call spacetime. This fabric of spacetime can be distorted by massive objects. Celestial objects such as planets, stars, blackholes, galaxies and etc. curves this fabric of spacetime, akin to how a bowling ball curves the rubber sheet it is placed on. Figure 1.1 visually shows how massive celetial objects may distort and curve the spacetime around it. With this description of spacetime curvature, it is possible to formulate

Figure 1.1. An artist’s rendition of a massive object curving the fabric of spacetime [1]. Einstein’s version of gravity. Einstein’s gravity is not a force, but rather curvature in spacetime created by the massive object defines the path that objects near it will follow. Figure 1.1 shows a smaller object following a path defined by the curvature in spacetime created by the more massive object. Using this defintion of gravity, let us run the the similar scenario again, where the sun vanishes. Once the sun has dissapeared, in Einstein’s theory of General Relativity, the planets would not immediately spin out of orbit. The removal of the

1. EINSTEIN’S CONCEPT OF GRAVITY

3

sun would essentially create a ripple in space time almost as if a pebble were dropped into a still water pond. This ripple will take the form of a wave that will travel at the speed of light, and take eight minutes to reach the earth. Thus, the gravitational effect of the sun dissapearing would not be instantaneous; it would be dependent on time. The more massive the object, the more it will curve spacetime and visa-versa. The greater the curvature of spacetime, the stronger the gravitational field that object will create. Thus, Einstein’s theory of gravity is unlike Newton’s where space and time are static; Einstein’s General Theory of relativity allows space and time to both be combined in one fabric and be manipulated by matter.

CHAPTER 2

Consequences of General Relativity The contradictions between Newtonian Gravity and Einstein’s Gravity led to a better understanding of fundamental physics. The interpertation that gravity is not a force but rather the path that matter follows in curved space time, has explained phenomena such as gravitational lensing, black holes and the expansion of our universe. This chapter will discuss these predictions of General Relativity in some depth. 1. Gravitional Lensing In his famous short article, “Lens-Like Action of a Star by the Deviation of Light in teh Gravitational Field,” Albert Einstein introduces the concept of Gravitational Lensing [2]. In this article, Einstein puts forth the idea that Gravitational Lensing is the bending of light around massive objects. This gravitational lens sometimes may produce multiple images of the distant object emmitting the light as it travels around the intervening massive object.

Figure 2.1. A drawing of light emitted by the source, getting bent by the intervening mass on its way to the observer. Matter traveling through curved spacetime will follow a path defined by that curvature. In Figure 2.1, the intervening mass curves the spacetime around it. As light nears this intervening mass, the photons’ path gets altered by this curved spacetime. 4

2. BLACK HOLES

5

This altered path is what we see as a “bending” of light. Moreover, this gravitational lens may also cause the observer to see multiple images of the same object. Figure 2.1 depicts a scenario that occurs sometimes: observing dual images; the light takes multiple paths to get to the observer as it passes the gravitational lens. If there are multiple paths of light for the same source, the observer may see multiple images of that source. The famous 1979 twin-quasar experiment served as evidence of gravitational lensing, when two images of a quasar were observed as its light passed by a lensing galaxy. 2. Black Holes Black holes are regions of very large amounts of mass, usually resulting from a gravitional collapse of a star. The mass of a black hole is so great that it creates a very deep, almost cone-like curvature in the fabric of spacetime.

Figures 2.2 and

Figure 2.2. An object with mass M curving spacetime [3]. 2.3 depict a scenario very similar to a gravitational collapse of an object such as a star. Figure 2.2 depicts an object with mass M curving the fabric of spacetime. During a gravitational collapse, the mass of the object stays the same, but its size decreases as its density increases. In the case of a black hole, the entire mass of the original object (before the gravitational collapse) gets focused into a singularity [4]. As seen in Figure 2.3, when the same mass is concentrated and more dense, it creates a deeper curvature in spacetime. In fact, spacetime around a black hole is so curved

2. BLACK HOLES

6

Figure 2.3. An object with the same mass M, but much smaller diameter, curving spacetime [3]. that light not only bends as it enters it (as seen in gravitational lensing), it cannot even escape the gravitational field of the black hole. The “blackness” of black holes results precisely from this phenomenon. Newtonian gravity states thats for very massive and dense objects, the escape velocity required to break free from their gravitational fields would be equal to the speed of light [5]. Going with this assumption, light should be able to escape the gravity of a black hole, as its speed is, well, equal to the speed of light. This is not the case, however, since light cannot escape a black hole. Einstein’s theory of General Relativty provides a more indepth reasoning for why light cannot escape the gravitational grips of a black hole, for which we must study the embedding diagrams of a black hole.

Far Away

Horizon Figure 2.4. Spatial Embedding Diagram [6].

2. BLACK HOLES

7

Figure 2.4 shows a spatial embedding diagram of a black hole. In the diagram, the top part is the structure of spacetime far away from the black hole, whereas lower (narrower) part is at a location very close to the event horizon of a black hole. Roughly speaking, the event horizon is the boundry of a black hole, which, if crossed, is the point of no return; if an observer passes the event horizon, they have no way of returning: a concept that will be discussed shortly. Note the relative flatness of spacetime far away from the black hole as opposed to the extreme curvature seen very close to the horizon of the black hole. Thus, as you get closer to a black hole, spacetime begins to curve deeply. To understand why light (or any object for that matter) cannot escape a black hole, however, we need to look at a different ebedding diagram. In Figure 2.5, the vertical lines are the time-like lines and the horizontal lines

6 4 2

T0 -2 -4 -6 -4

-6 -6

-2 0

-4

2

-2

4

0

X

6

2

Y

8

4 6

10

Figure 2.5. Space-time Embedding Diagram [6]. are the space-like lines. Figure 2.6 shows a bird’s eye view of a spacetime embedding diagram. At either ends of the diagram, labeled as the flanges, the observer is far away from the black hole. Spacetime is very flat at the flanges. However, as the observer moves away from the flanges, toward Y = X = T = 0, spacetime begins to

2. BLACK HOLES

8

0

Flange T0

Flange X

Y

0 +20

Figure 2.6. Bird’s Eye view of Spacetime Embedding Diagram [6].

curve heavily. If the observer is approaching the singularity of the black hole, they will need to essentially climb out of the black hole’s spacetime curvature by moving towards the flanges of the embedding diagram. However, this poses the question: why are observors, even those traveling at the speed of light, unable to move back out to the flanges of the embedding diagram? The answer to this question lies in the two separate timelike and spacelike lines in the embedding diagram. While we are far away from the black hole, in the flanges shown in Figure 2.6, the timelike paths do not lead to the center of the black hole. As such, the obsever is free to travel anywhere along those timelike lines (note, however, as a rule of General Relativity, the observer is only free to travel forward in time, not backwards). However, notice that as you enter the curved space (shown by the curving spacelike lines near the horizon of the black hole in Figure 2.6) the timelike lines also start becoming distorted. Along with the spacelike paths, the timelike paths also start tilting towards the center of the black hole’s mass as the observer approaches the event horizon. As the distance between the center of mass

3. THE EXPANDING UNIVERSE

9

and the observer decreases, the distortion in the timelike paths becomes more and more pronounced [4]. Figure 2.6 illustrates that exceedingly many timelike paths lead to the center of mass as the observer moves away from the flanges and towards the black hole. Once the observer has reached the event horizon of the black hole, all timelike paths lead to the center of the black hole’s mass. Since the observer cannot move backwards on these timelike paths, they must move forward along the paths, but all possible paths lead to the center of mass. As such, once the observer (whether it be light or any other object) has reached the event horizon they are forever trapped in the black hole because they have no timelike paths available to them that lead outside the black hole. Thus, the concept of escape velocity in Newtonian gravity only gets us half-way in understanding why an object cannot escape the gravitational field of a black hole. The inherent contradiction of an object traveling at the speed of light not being to escape a black hole is only answered through General Relativity with the help of timelike paths, spacelike paths and their curvatures.

3. The Expanding Universe For years scientists have wondered about the ultimate fate of the universe. With the help of General Relativity, they came one step closer to answering that question. Upon observing other galaxies in the universe, physicists discovered that their light is redshifted [5]. That is, the wavelenghts of the light coming from those galaxies are bigger than they are in nearby galaxies, indicating that they are moving away from us. Now, this could merely mean that the galaxies were moviing away from us, we were the center of the universe and the universe was not expanding. However, Hubble’s Law, given by, (2.1)

V = H0 d,

3. THE EXPANDING UNIVERSE

10

gives light to the idea that all galaxies are moving away from each other at the same speed. In Hubble’s law, H0 is Hubble’s constant, and V is the velocity of the receding galaxy, which is related to the shift in wavelength (2.2)

∆λ λ

by the Doppler relation,

V ∆λ = ≡ z, c λ

where z is the cosmological redshift. In Equation (2.1), it is the observation that H0 is constant that vindicates that our universe is expanding. Hubble’s constant can be given by the relation, (2.3)

H0 ≡ H(t0 ) ≡

a(t ˙ 0) , a(t0 )

where a(t) is the scale factor that represents the relative expansion of the universe and it is also a function of time. As such, specifically, a(t0 ) is the scale factor at the time the light signal is recieved from another galaxy. Observation has shown that regardless of the distance between the two galaxies,

a(t ˙ 0) a(t0 )

is a constant of 72 ± 7 (km/s) . M pc

This essentially means that the change in the scale factor during a given time divided by the scale factor at the time of reception is always constant, indicating that the universe is expanding uniformally. Moreover, a relation that combines Equations (2.2) and (2.3), gives further insight into the expansion of the universe: (2.4)

z≡

∆λ a(te ) = . λ a(t0 )

3. THE EXPANDING UNIVERSE

11

Simplifying, ∆λ λ λ0 − λe z = λe λ0 z = −1 λe λ0 . 1+z = λe

(2.5)

z =

(2.6) (2.7) (2.8)

This simplification allows us to reform Equation (2.4) as, (2.9)

1+z ≡

λ0 a(te ) . = λe a(t0 )

The observation that light coming from other galaxies is redshifted implies that wavelengths are becoming longer; a(te ) a(t0 )

λ0 λe

will be greater than one. This then suggests that

> 1, which allows us to infer that the relative expansion of the universe at

the time of reception, a(t0 ), is larger than the relative expansion of the unvierse at the time of emission, a(te ). In less mathematical terms, the the distance between the galaxies grows larger over time, which vindicates the theory that our universe is expanding. Going a bit deeper, we can understand the different possibilities of spacetime geometries for our universe. While the metric, (2.10)

  ds2 = −dt2 + a2 (t) dr2 + r2 (dθ2 + sin2 θdφ2 ) ,

given by physicists Friedmann, Robertson and Walker is for a flat universe, a more general version of this metric is given by: (2.11)

 dr2 2 2 2 2 + r (dθ + sin θdφ ) . ds = −dt + a (t) 1 − kr2 2

2

2



3. THE EXPANDING UNIVERSE

12

The general FRW metric gives room for different spacetime gemoetries for our universe: closed, open or flat. Figure 2.7 shows visually what we ...