Introduction

It is often difficult to speak about things like singularities because of their prevalence in pop culture. Oftentimes a concept like this takes a life of its own, forever ingrained in ones imagination as a still from a movie (for me this is that scene from Inception where they encounter Gargantua for the first time). Like many concepts in theoretical physics, popular culture is often better at bringing them into light than it is at bringing them into focus. In this article I will try to explain in simple terms what a singularity is and how that relates to physical reality. As always, I will give an exact example of the singularity by means of an equation. At the end, once the mathematics is clear, I will try to explain what the physical reality of the singularity is.

Mathematical Singularities

Singularity of

1. Behavior of the Function:

- As (approaching from the positive side): - As (approaching from the negative side):

At , the function becomes infinitely large (or small), making a singularity. This is a pole of the function where the value tends to infinity.

2. Undefined at the Singularity:

The function is undefined at , which is the point of discontinuity.

In mathematics, singularities are not a problem.

Physics Singularities

The singularity of a black hole can be described by the Schwarzschild metric, which is the solution to Einstein’s field equations for a non-rotating, uncharged black hole. The Schwarzschild metric is given by:

Where:

• is the spacetime interval,

• is the speed of light,

• is the gravitational constant,

• is the mass of the black hole,

• is the radial coordinate,

• and are angular coordinates.

Be careful though these are not polar co-ordinates, these are coordinates for the Schwarzschild metric. They are a kind of nested spherical coordinate system, this does not seem to affect the solution but helpful to know.

The singularity occurs at . As , the term grows without bound, leading to an infinite curvature of spacetime. This represents the physical singularity of the black hole.

Additionally, the component of the Schwarzschild metric, which is the time-time component, becomes singular as :

As , , indicating the breakdown of spacetime and the presence of a singularity.

You can create another singularity by setting in the metric, this is the event horizon of the black hole. This is the point at which light can no longer escape the black hole. However, this is solely a mathematical singularity, since you can still define the metric at this point by a change of coordinates. One such set of coordinates is the Kruskal-Szekeres coordinates, which are used to describe the Schwarzschild metric in a way that is regular across the event horizon.

The Schwarzschild metric in Kruskal-Szekeres coordinates is given by:

where is a function of and , implicitly determined by:

Here, is the Schwarzschild radius:

The coordinate singularity at in the Schwarzschild metric is removed by transforming to Kruskal-Szekeres coordinates, and the metric remains regular across the event horizon.

Another Physics Singularity

Again, starting from yet another solution for the field equations we can derive FLRW metric (Friedmann-Lemaître-Robertson-Walker metric) which describes the universe as a whole. The words homogenous and isotropic, effectively mean that instead of considering each individual planet in the universe as an actual individual body, we consider them to be individual particles in a fluid (in fact, the FLRW metric considers each galaxy to be a particle). We do this so that we can use equations for fluids to simplify the stress energy tensor in the Field Equations. Our strategy to solve the field equations is as follows,

1. Assume the universe is some kind of fluid (so basically zoom out till all the galaxies look like a fluid)
2. From 1, you can write down the stress energy tensor for the fluid, this is a simple equation (This is for the Schwarzschild metric, and for many other useful metrics, so we never really had this problem before, but when you zoom out you need it)

The FLRW metric, which describes a homogeneous and isotropic universe, is given by:

Where:

• is the spacetime interval,
• is the speed of light,
• is the cosmic time,
• is the scale factor of the universe,
• is the radial coordinate,
• and are angular coordinates,
• is the curvature of space, which can be , , or .
• The scale factor describes how the universe expands or contracts with time.
• The curvature parameter determines the geometry of space: negative curvature for , flat curvature for , and positive curvature for .

Friedmann Equations Recap

The Big Bang is represented in the Friedmann equations as a singularity at the beginning of time when the scale factor approaches zero. This signifies an initial state of infinite density, temperature, and curvature.

The Friedmann equations in cosmology are derived from Einstein’s field equations for a homogeneous and isotropic universe. Assuming zero cosmological constant (), they are:

1. First Friedmann Equation:

2. Second Friedmann Equation (acceleration equation):

3. Continuity Equation (conservation of energy):

where: - is the scale factor (the “size” of the universe at a given time ), - $ \rho {=tex}$ is the energy density, - is the pressure, - is the gravitational constant, - is the curvature parameter ( for a flat universe, for closed, and for open).

Representation of the Big Bang Singularity

In the context of the Friedmann equations, the Big Bang is identified by the conditions: - $ a(t) \to 0{=tex} t \to 0{=tex} \rho {=tex}\to {=tex}\infty {=tex} a(t) \to 0{=tex} $ (implying infinite density and temperature), - Curvature becomes infinite, signaling a physical singularity.

Explanation Using the First Friedmann Equation

In the first Friedmann equation:

As $ t \to 0{=tex} $: - The scale factor approaches zero. - For a positive energy density $ \rho {=tex} \frac{\dot{a}}{a}{=tex} $ (known as the Hubble parameter) goes to infinity, meaning the rate of expansion is initially unbounded. - If $ a \to 0{=tex} \rho {=tex}\to {=tex}\infty {=tex} \rho {=tex}$ is inversely related to the volume of the universe.

Thus, at , the universe is in a state of infinite density and infinite curvature, which we identify as the Big Bang singularity.

Continuity Equation and Energy Conservation

The continuity equation:

implies that as approaches zero, the rapid change in the scale factor causes the energy density $ \rho {=tex}$ to increase sharply, reinforcing the singularity concept.

Physical Interpretation

At , when the scale factor , the energy density $ \rho {=tex}$ theoretically becomes infinite, meaning all mass, energy, and curvature are compressed into a single point. This condition marks the beginning of the universe, as described by the Big Bang theory, before which classical descriptions of time and space may no longer apply due to quantum gravitational effects.

In short, the Big Bang singularity in the Friedmann equations marks the initial state of the universe at , where , density and temperature are infinite, and classical general relativity predicts a breakdown in spacetime structure.

Connection to Reality

While all of the above can be found in a basic undergraduate textbook, I think the goal of me writing this post was to have a collection of examples of singularities both from mathematics and physics to reinforce the idea of reality. While in the mathematical examples, the does not represent an actual place that we can go and take measurements of , but what if we did? What if we indeed knew a physical place in the world, where the function really described the behavior of the world. This is not hard, you could imagine this as the share that each person gets (of a cake or similarly sweet treat) if there are people. If there are people, each person gets of a share. What does it mean to have people? This is the kind of question that the mathematical singularity is trying to answer. But it is physically impossible to have people, so the singularity is not a real place. If you had people and a cake, the question of dividing it does not make sense. In much the same way, the singularity of the Schwarzschild metric is not a real place, it is a place where the equations break down. This does not mean that some wild stuff happens at the singularity, it means that the equations we are using to describe the world are not valid at that point. This is the same as saying that the function is not defined at . Very often in movies, the singularity is portrayed as a place where the laws of physics break down. This is not true it is just that the laws of physics defined by the equations work everywhere else but not at that point. This could mean one of two things, 1. The equations are not valid at that point, so we need to find new equations that are valid at that point. 2. Some wild stuff happens at that point, and we need to find out what that is. And rework our equations to include that.

But simply by looking at the equations, we cannot say which of the two is true. We need to go out and measure the world to find out.

References

https://diposit.ub.edu/dspace/bitstream/2445/59759/1/TFG-Arnau-Romeu-Joan.pdf

Introduction

Like most people I was taught the field equations for Newtonian gravity in high school in order to prepare for the national selection exams. I went on to study mathematics and differential equations and learned more field equations for voltage etc. Since then, I have always read, mostly in popular science books, that the Einstein field equations are ‘’better’’ than Newton’s. Popular science books are great for getting a general idea of what very complex concepts but all too often they end up being a collection of mismatched metaphors and comparisons (think of Einstein’s rubber sheet preceded by Newton’s Apple) that often leave one wondering what mathematical framework could possibly juxtapose the two. As always, I try to create a minimum working example that would help me (you, us) understand the problem better. In this post I will compare the two theories using the example of Mars’ orbit.
Before we begin, this essay was never meant to be an exhaustive explanation of the math, rather it was meant to be a reasonably sophisticated roadmap that would allow one to see an equation in action and then intelligently Google it’s origins. With that end, starting from the Schwarzschild metric is a good point to start from, since you only need differential equations from that point on to solve things. I have skipped several steps from the Schwarzschild metric to the final differential equation, but most of those steps are just chugging through the math, they do not really help you understand the problem. Basically, my ‘recipe’ is as follows :

• you have a system of equations (Einstein’s field equations)

• you have a solution to that system (Schwarzschild metric) that makes some simplifying assumptions on the constants

• you plug in values for the constants and solve the differential equations for your system (Mars’ orbit)

You can use this recipe for any field equations, not just gravity. And indeed for any other field theory of physics.

General Relativity

Starting from the first principles of Newtonian gravity one can solve field equations for gravity that yield the following solution

Where is the distance of a planet from a star. The general solution looks like . From which you can generate perfectly elliptical orbits as well as those that are not. Recall that Newton did not provide a reason for why masses attracted each other. Einstein proposed that mass warped space-time and this caused gravity as a secondary effect, which is why his field equations are not directly comparable to Newton’s but in summary one can choose solutions based on some symmetrical conditions, since we are looking at a large spherical star we can choose the Scwarzchild metric,

All that finally gives us an equation like the one for Newton,

Which can be written as , For comparison here is Newton’s equation,

Orbits are perfectly elliptical in the first case i.e. every year you would expect Mars to be in the same place. But Einstein’s equations give a spiralling orbit such that every year Mars ends up moved or slightly precessed. It is exactly this phenomenon, called the precession of Mars, that proves that Einstein’s equations are considered a better explanation for gravity than Newton’s.

Conclusion

Well, that is all there is to it. Reasonable next steps for the interested reader would involve some combination of the following :

• Understand the origins of Einstein’s field equations (this can get quite philosophical)
• Understand the Schwarzschild metric and how it is derived as a solution to the field equations
• Understand the precession of Mars and how it is used to validate Einstein’s field equations via the error in the orbit of Mars as predicted by Newton’s equations
• (extra credit) Understand the Schwarzschild metric in the context of the Kerr metric (rotating black holes) and the Reissner-Nordstrom metric (charged black holes)

Addendum : What does “choosing a solution” to the Einstein metric mean?

Infuriatingly, often one hears of Einsteins equations but no one really ever shows you what they are. Here is a brief overview of the primary concepts. Every equation is a tensor equation, which means that it is a set of equations that are invariant under a change of coordinates. You can find them in the non tensor form too, no you dont need to know what tensors are to understand them. Einstein’s field equations are a set of ten interrelated differential equations in Albert Einstein’s general theory of relativity. They describe how mass and energy in the universe influence the curvature of space-time, which in turn affects the motion of objects. The equations can be summarized as (in tensor notation)

Here, represents the Einstein tensor, which describes the curvature of space-time, and is the stress-energy tensor, representing the distribution of mass, energy, and momentum in space-time.

The equations essentially state that the curvature of space-time
(left-hand side) is directly related to the energy and momentum of matter and radiation (right-hand side).

This formulation has profound implications for our understanding of gravity, leading to predictions such as the bending of light by massive objects, the slowing of time in gravitational fields, and the existence of black holes.

Choosing a solution to Einstein’s field equations involves finding solutions that satisfy both the mathematical requirements of the equations and physical constraints. The Schwarzschild metric is one such solution that describes the geometry of space-time outside a spherically symmetric, non-rotating mass, such as a black hole or a massive celestial body (literally 90% of the solutions you ever need are this case, charged black holes are still an open problem in almost all modern equations, other metrics exist but for the most part you only need this one to get an idea). The Schwarzschild metric is given by:

Where: - is the spacetime interval, - is the time interval, - is the radial interval, - is the solid angle element, - is the gravitational constant, - is the mass of the central object, - is the speed of light, and - is the radial coordinate.

The Schwarzschild metric satisfies Einstein’s field equations for vacuum solutions (i.e., regions of space-time where there is no matter or energy), which means . The RHS of the equation above goes to zero. This solution describes the curvature of space-time around a spherically symmetric mass. It exhibits features such as gravitational time dilation, gravitational redshift, and the existence of an event horizon for , beyond which no information can escape (defining the boundary of a black hole).

The Schwarzschild metric was initially derived for the case of a non-rotating, uncharged mass. However, it serves as a fundamental solution in general relativity and has been foundational in understanding the behavior of space-time around massive objects.

References

• Schwarzschild metric
• Einstein field equations
• General relativity
• Schwarzschild black hole
• Excellent channel : https://www.youtube.com/@ScienceClicEN
• Derive the Einstein tensor : https://www.youtube.com/watch?v=4g1xZNKw2cc
• BUT WHAT IS A TENSOR : https://www.youtube.com/watch?v=Hf-BxbtCg_A
• Difference in orbitals, full proof : https://nhsjs.com/2023/the-difference-in-orbits-of-planets-in-terms-of-newtonian-mechanics-and-einsteinian-mechanics/