General Relativity 101


Screenshot of the conversation
that instigated this post.
The Credible Hulk made a few posts on General Relativity, which is Einstein's theory of gravity. The comment thread to the right (image) ensued.

I have taught myself some general relativity, and I am aiming for a Ph. D. that involves general relativity. Can I explain it, even if slightly?

I think I can, at least, try to do so. So, here I will try to explain the concepts involved:
  • Riemann Tensor
  • Ricci Scalar
  • Ricci Tensor
  • Christoffel Symbol
  • Energy-Momentum Tensor
  • .. 
As you can see, general relativity is scary. It really is - one of the standard texts for an introduction course is Spacetime and Geometry by Sean Carroll, and the first four chapters are mathematics. 

Yet the information contained does have conceptual explanations. There's a familiar one, involving space as a sort of elastic cloth and planets and stars as really heavy balls that dent the fabric. 

However, I also want to provide some concept for the mathematical objects involved. That's the beef of this article.

Let's start with a movie. Had I mentioned it's been a hundred years since this theory was cooked up?

How to scare your readers away

So this is the section where people would typically flee. Mathematics was scary in high school, frightening in university and is just plain horrifying when you've left school.

As a physicist, I no longer see mathematics as something scary. It is a language, and like any language it describes objects and interactions. Instead of focussing on the math, we'll focus on the concepts involved. I will give them their proper name, that is, the mathematical notation.

An ant is confused on top of a hot
air balloon. How did I get here?
How did I get so big?
Let's start with a very, very brief history. Suppose you're an ant on a  balloon. Not a big ant on a big balloon as in the righthand image, but a very small ant on a very big balloon. Can you figure out that it's a balloon? How? You can only see the horizon, and everything seems flat; much like a human on the earth.

Turns out you can. A guy named Gauss figured out that you could, and wrote some reasonably sophisticated mathematics to say just that: You only need the surface to find out that it's curved. 

Following that, several students, including Riemann got to work. Riemann figured out how to generalise the notion above to an arbitrary amount of dimensions. That gives us the first name, and the first object.

The Riemann Tensor, mathematical notion $R^\lambda_{\:\sigma\mu\nu}$, is a mathematical object that contains all the information about a certain space. So, if you want to describe something happening on a surface shaped like a sphere, then the Riemann Tensor contains that information. The object consists of elements. To visualise that, imagine a drawer. You pull it out, and on its sides there's more drawers. And you pull one of those out, and it contains another drawer. Each drawer is numbered. So, if you're asked to look for the element 2134, you pull the second drawer. Of that drawer, you pull the first; of that one, the third. Finally, the fourth; and that is where this element of the information is. In what we will discuss, we have the three dimensions of space and one of time, so that there are four dimensions in total.

The Riemann Tensor is defined purely in terms of another mathematical object, called the Christoffel Symbol, notation $\Gamma^\lambda_{\mu\nu}$. This one is a symbol, not a tensor. This remark might seem superfluous, but it is important to make the distinction even if the meaning escapes you. So, it is a symbol, not a tensor.

The Riemann Tensor has $4\cdot4\cdot4\cdot4$ elements. That's a lot of elements. What if we want a physically relevant object that is a lot smaller? Well, there's two candidates that are relevant to physics. The first is the Ricci Tensor, $R_{\mu\nu}$, which only has $16$ elements. The Ricci Tensor encodes some information of the Riemann Tensor, but not all of it. It is really the smaller physically relevant object. And from it, we can find the Ricci Scalar, also known as the curvature scalar, notation $R$. This is a number, the simplest mathematical object.


You now know that the Riemann Tensor tells you everything about a curved space, while the Ricci tensor and scalar are just smaller objects that contain only a part of that information. I've also told you that the Riemann Tensor is defined in terms of the Christoffel symbol, but I didn't tell you what that means.

The christoffel symbol is what leads to what we think of as gravity. There is an equation, called the geodesic equation, that explains this connection. The equation is for light or other massless objects, and really just denotes the shortest path. However, there might be constraints.

Consider two points on a sphere. What is the shortest path? Is it the blue path, path A, or is it the green path, path B? Both of us know that path A is the shortest path. However, that is if you're free to move. An ant on this sphere can only choose path B, which is the shortest path if you are constrained to the sphere. 

In physics, as you might remember, a lot of things move along the past of the least resistance. This is true for all manner of things, and its origins lie in the principle of minimisation of energy. So, the constraint that the ant walks on the spherical surface can probably be realised by a resistance, by a force.

What is that force? Well, for curved space, there is a well known name attached to this resistance. It's called gravity. We have just described how it's not really a force, but a result of the surface, of the space, being curved. Gravity is the fictitious force felt because space is curved.

But how does all the weird mathematics come into this? Well, the christoffel symbol is involved in the geodesic equation, which is one of two equations that we will describe:
$$\begin{align} \frac{d^2x^\mu}{d\lambda^2} &= - \Gamma^\mu_{\rho\sigma} \frac{d x^\rho}{d\lambda}\frac{d x^\sigma}{d\lambda}\end{align}$$

Light is bent by gravity, an effect that serves as
empirical evidence and is used in astronomy
to observe far-away objects.
The term on the left is just the acceleration. Newton says that mass times acceleration equals force, so the term on the right is the force; gravity. This tells you that light, which moves along geodesics, is bent by gravity. This effect is used in astronomy, and serves as part of the experimental evidence for general relativity. 

Mind you, the entire list of objects above is zero for space that is flat. So, in flat space, a geodesic is a straight line. This is why it took so long to think of general relativity; we could not locally find any evidence of curvature. 

In space that is almost flat, we can use physicists favourite trick and use perturbation theory. Perturbation theory can greatly simplify complicated problems. For instance, the simple solutions for a music string are found using perturbation theory. The idea is that, because the space is almost flat, we can take the flat-space solution and then try solutions that are only slightly different.

Guess what that gives you in almost flat space? The answer is, of course, Newton's description of gravity. This should come as no surprise; we live in almost flat space, so the minimum requirement for the theory is that it reproduces those results. 

There is one object that I've neglected to mention. It is called the metric tensor, and it is actually more fundamental than the Christoffel symbol. The latter is in fact derived from the metric tensor. There are a number of ways to think about the metric tensor. The one that I find most striking is that the metric tensor tells you how time passes for an observer on the object, which is called proper time. Additionally, the metric tells you about angles, areas and volumes. 

Now, we are ready to write down Einstein's field equation:
$$\begin{align} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} &= 8 \pi G T_{\mu\nu}\end{align}$$

These are called the Einstein Field Equations. But what does it tell you? Well, the objects on the left are the Ricci Tensor, Ricci scalar, the metric and the cosmological constant $\Lambda$. Interestingly, Einstein originally thought that including the latter was his biggest mistake, but more recent research on dark energy/matter has made it clear that he was correct. Some interesting communication between Einstein and Schrödinger even hinted at this.  All of these have to do with curvature. On the right, we have the energy-momentum tensor times the gravitational constant $G$, which describes the strength of gravity. This object is pretty close to its name; it is a mathematical object with components that describe how matter or energy is distributed and how it flows. Two of the fairly common examples are dust, which has no pressure, and a perfect fluid, which does. 

But that is an astounding equation; it tells you that mass shapes spacetime. And spacetime influenced the motion of that mass, which shapes spacetime. We can immediately see that solutions in general relativity are hard - the interaction never ends! 

I've always found the model of light bending around heavy objects as the most helpful mental tool. It allows you to imagine a thin line of light bending, thus mapping the space for you.

If I'm not mistaken, then with some difficulty you are now able to identify the components of the Einstein Field Equations. What's more, you know that this tells you how the metric changes, which tells you how the christoffel symbol changes, which describes gravity.

Before we finish, I'd like to discuss two more things. The first is one of the eldest results, the orbit of Mercury. The second is a recent result, the discovery of gravitational waves.

Schwarzschild and Mercury

Mercury's advancing perihelion.
Mercury's orbit is not an ellipse,
as Newton's gravity predicts, but
is a flower petal shape - as
Einstein  successfully explained.
Source: Scientific Explorer
One of the first symmetric objects you can imagine is probably a sphere. A sphere is maximally symmetric; it looks the same from all sides. A curved space can have a similar property, and this leads to the notion of the Schwarzschild solution. This is "just" the perfectly spherical solution to Einstein's Field equations . It also leads to a prediction of black holes.

Consider the gravitational field of the sun for an object that comes reasonably close. That object travels through vacuum and we assume it is not heavy enough to affect the sun in turn.

As a result, the effective gravitational potential includes terms of a higher order in the distance from the sun than it does for Newton's gravity. This is most pronounced for mercury, because it is both the closest to the sun and comes very close at one time in its orbit. It probably no longer surprises you when I tell you that mercury's orbit did not agree well with Newton and that Einstein did completely explain its orbit. See also the picture to the right.

Gravitational waves

If we turn to the physicists favourite trick, perturbation theory, and apply it to Einstein's Field Equations, we would find that Einstein's equations are wave equations. Wave equations are a class of equations that are similar to waves; as a result, they allow wave solutions.

Which means that Einstein's gravity predicts gravitational waves. Note that gravity waves are something completely different, and have to do with atmospheric waves. But how to measure them?

Well, as you probably know, physicists at LIGO have succeeded in that measurements, and it would be a shame not to include one of their promotional videos.

That's awesome. If someone asks you what a gravitational wave is, then first answer that its a ripple in space time. It sounds awesome, but is still vague. If they continue asking, speak about a periodic bending of light away from straight lines.


Let's consider what we've described. We started with a vague notion of curvature, and discovered that this information is in the Riemann Tensor. We described smaller, less informative objects called the Ricci tensor and Ricci scalar. We spoke about how gravity bends light, by a certain amount described by the Christoffel symbol, which also happens to describe gravity. We spoke about the metric, and how it defines the Christoffel Symbols. In short, we described a load of mathematical, scary terms that somehow have to do with Physics.

If I've succeeded at my purpose, there is now some concept for each entity in Einstein's Field Equations and of gravitational waves. That'll have to do; Master courses in general relativity take about 180 hours, which isn't enough to do anything useful with it.  

Norway shows no scientific evidence of GM food safety


In a comment to a pro-GM post, someone served me this post "No scientific evidence of GM food safety".

That sounds suspect, but it is a post about a report by the Norwegian Environment Agency. So what's going on?

The first thing that sprang to mind is the difference between the following two statements:
  • GM products are safe, because there is no evidence of harm.
  • GM products are not unsafe, because there is no evidence of harm.
The first can be false while the second is true. Among others, the first hypothesis requires far more evidence than the second. 

The report itself is titled "Sustainability Assessment of Genetically Modified Herbicide Tolerant Crops", subtitled "The case of Intacta Roundup Ready 2Pro Soybean Farming in Brazil in light of the Norwegian Gene Technology Act". This is a case study, not a study of GM in general. The report is in two parts, the first addressing the ecological aspects of the GM plant and its related herbicide. The second part, which we will not consider here, regards social and economic issues.

Note that sustainability in the report title refers to the possibility of people to satisfy their needs without compromising the needs fulfillment of future generations, a notion I fully support.Norwegian biosafety legislation is formulated in terms of the precautionary principle, which means that there is a certain amount of subjectivity involved. In principle, the deciding measure is the scientific uncertainty concerning absence of harm and absence of a mechanism for harm, but the latter is often ignored.

Finally, note that regardless of this report in Norway, Brazil has approved for open-field production, human consumption and animal feed.

The report

A compilation of the first five pages of the report. Note that
the source of the underlying image isn't too interesting. The
image shows a comparison between the Intacta line and a
Roundup Ready line, thus displaying the effect of the
resistance tot he Lepidopteran Insects.
The crop under consideration is resistant to both the herbicide glyphosate and Lepidopteran insects, which means moths, butterflies and thus their larvae. The sources of information are a dossier submitted by Monsanto to the Brazilian National Biosafety Technical Commission. Furthermore, they consult relevant literature found by various searching tools.

The Intacta Soybean is a variety developed by Monsanto, resulting from a crossbreed of two GM parental lines that express resistance to one, but not both, sources.

I compiled an image compiling the first five pages, which can be found to the right. At this point, most of the disagreement between the report and Monsanto is in minor details, differences between Continental American and EFSA guidelines. Also note that (right side) they compared the nutritional profile of Intacta to an organic counterpart instead of the conventional non-GM soy in the same study. The write that this is contrary to Monsanto's files, but organic farming generally implies a different scheme that does not focus on yield or economic feasibility. It is thus not surprising that this profile is different.

In the second part, the interaction between Intacta soy and the environment, the first guiding question is not answered. It is important to note that this is an application made to Brazil which is evaluated post-hoc by a Norwegian agency; as a result, the required information is not the same. Based solely on this difference, the report can deem Intacta soy as not sufficiently safe for Norwegian use,
and the study could be used to propel anti-GM propaganda.

The report reiterates the point that combined analysis of different agroecosystems is not appropriate. Here, they explain that this is because difference resulting from the interaction between agroecosystem and the plant may be masked by data aggregation. However, they do not report if the data shows any such tendency nor do they give an example of such interaction in relevant literature. 

They discuss a study that shows differences in gene expression and protein profiles between GM and non-GM varieties in different studies, a report that I find lacking in statistics, since it focusses on principal component analysis without addressing statistical significance. Either way, it is unclear if the EFSA guidelines require such an analysis. Given that the report has the intent to show how future evaluations should go, I think they included it as a point of interest and perhaps something warranting investigation if it should be included. I do not know to what extent, if at all, the GM maize agrosystem variation can be generalised.

A rather odd discussion follows. The first state that Monsanto reports similar phenotypical development related to plant metabolism, similar chemical composition and nutritional value at different sites, compared to the GM parental lines and an unspecified conventional control.

They then note several studies that found that different soil types and glyphosate rates can lead to different nutrient content and dry matter, the latter for higher rates. This is not weird; resistance isn't immunity, so the herbicide does affect them. However, they then assert that this indicates that changes in gene expression and plant characteristics. I think this is a leap of faith, as higher rates of glyphosate generally indicates the presence of weeds that take away from the nutrition content in the soil. As a result, you would expect that even after the weeds are killed off, they have still taken nutrition from the ground that cannot end up in the crop. I also find issue with their formulation, which indicates that Monsanto did not address such considerations despite EFSA specifications. However, EFSA was not a factor for Monsanto, as it was applying for approvement in Brazil. No difference in ecological interactions between Intacta soybean, parent lines and conventional soybean were found.

Soybean is a self-fertilizing plant, so that there are only low levels of cross-pollination. Vertical gene-flow, essentially having a GM soybean and a non-GM soybean in neighbouring fields with these possibly fertilizing each other, is nearly non-existent and should be of no concern. Even more so as seed-saving is not common practise in modern agriculture. Horizontal gene transfer, which is between different species, is dismissed by both the report and Monsanto. The report does add a footnote to that, but does not specify why this is a concern.

Volunteer plants, which I've interpreted as leftover seeds or plants that survive the harvesting cycle, thus effectively being a weed in the next harvest, seem to be a very low risk. As soybeans are beans, the spread of such seeds is low and the seeds are that which is harvested in any case. The introduction of GM traits is cited to not alter the germination potential or confer competitive advantage. For clarity, this regards the number of seeds or the chances of producing seeds.

Now, question of safety is addressed, specifically on non-target organisms. The Intacta soybean was developed to control a broad spectrum of weeds and Lepidopteran insect pests. Monsanto did field tests in two locations on non-target insects, including arthropods, and reports no difference observed in terms of the abundance and distribution of insects except for target insect pests. Field tests with a parent line have also included mice, birds, soil decomposers and beneficial insects, including honey bee and spotted lady beetle. Again, no adverse impacts were found. This field test is relevant because we had previously established no chemical difference between the two parent GM lines.

The report continues on by noting the literature also reports no adverse effects on biological diversity due to either Intacta or similar GM soybean, or even in general for GM crops. Finally, they note some glyphosate-related issues on biodiversity at landscape and species level, but these are related more to glyphosate spraying practises and the liberation of farmland than to this specific crop. These are issues of farming practise and a supposed lack of regulations regarding pesticide appliance.

Next, we turn to what I expect to be the linchpin of the non-sustainability verdict. Monsanto did not compare with the closest non-GM soybean in all trials. Each trial has a specific, singular, purpose and only a few included conventional soybean. This is a point of major concern for the report: "The applicant (Monsanto) fails to distinguish among the following genetic material: (i) Intacta RR2 Pro, (ii) its GM parental lines, (iii) the conventional cultivars used for developing the GM parental lines". This is again formulated as a failure, instead of a result of the application being for Brazil, not EFSA. They also note, again, that combined analysis is used for the different test sites, rather than analysis for each site respectively. Additionally, Monsanto did not properly report whether these results are under real field conditions. This is more a question of documentation and methodology with regards to different guidelines than it is to safety.

This concludes the content of the report that regards environmental sustainability and ecology of the Intacta Soybean (pages 6-23). Note that the report contains an equal part about glyphosate (pages 23-50) and a larger part about socioeconomic sustainability (pages 50-84). 


The report essentially concludes that the sustainability of Intacta Soy and its related pesticide do not sufficiently satisfy the safety criteria. The reasons for this are mostly related to differences in the guidelines of Brazil and EFSA, and do not regard the evidence. Furthermore, several studies are pulled into the discussion that are of limited relevance.

As expected, it is extremely important to differentiate between the question of safety and the question of unsafety. The title of the blog posted served to me, No scientific evidence of GM food safety, is taking extreme liberties in formulation. A more accurate title would be insufficient evidence in an application in Brazil to satisfy EFSA guidelines

I find it of particular interest that the report found  it necessary to include studies that seem to have either little relevance or only have tentative conclusions. A far more reasonable stance, representing scientific consensus: No scientific research so far has detected significant hazards connected with the use of GE crops, despite the debate still being intense (source). 

Autism Day


Today is Autism Day. Autism is an often misunderstood diagnosis; many think of it as a disease. Furthermore, there is a thriving industry of autism woo that capitalises on that misunderstanding.

Many people have some traits, but not nearly enough to be diagnosed. I will explain this as I discuss the biological origins of autism. If you are interested to see if you have some traits, the AQ test is an amusing start.I score 43 on that test, which is not that weird as I have an official diagnosis of Autism Spectrum Disorder (ASD).

What is Autism? 

Wikipedia describes the characteristics rather well:
Under the DSM-5, autism is characterized by persistent deficits in social communication and interaction across multiple contexts, as well as restricted, repetitive patterns of behavior, interests, or activities. These deficits are present in early childhood, and lead to clinically significant functional impairment. 
Vineland score, adapted from BioRxiv.
Deficit means that it is less than some baseline, in this context probably that expected of a person of that age.  Clinically significant functional impairment is a rather strong and confusing term. It basically means that the deficit causes problems for which you (might) want to seek help.

In a recent article in Nature Genetics (NG), many interesting things were noted. For instance, they describe a Vineland scale of adaptive behaviour that gives a measure of social, communication and daily living skills as reported by the parents of 2497 diagnosed children and their 1861 siblings as a control. A histogram of the data can be seen to the right, where there is a very clear difference between the scores of the two groups.

But what are its origins? As the NG article shows, those genetic components that often coincide with an autism diagnosis are spread among the population. The degree of heritability is also very high, as shown by twin studies. More evidence for the genetic component being primary lies in mouse and even macaque studies. As many as fifteen genes are associated with autism; this is just the start. For accuracy, I must inform you that the genetic component does not fully explain autism.

There is a very interesting and helpful perspective called neurodiversity. As the NG article shows, traits are common throughout the population. Everyone is different, unique. Many other traits, e.g. length, intelligence, ability to keep up with technology, are variable within a population. Is that not normal biological variation (diversity)? Brains are biological; their growth is controlled by genes and genes vary by many different processes. Variation is to be expected. Autism can be seen as a group identified by a (complex) trait, just as you can identify the jocks and cheerleaders in any American High School movie. The decision of the DSM-5 (2013) to speak of ASD instead of a few subtypes reflects this view. That decision was based on genetic and neurological studies.

The actual traits of autism, despite the diagnostic criteria, are differently felt by those with a diagnosis. Instead of focussing on the diagnosis, I will just tell some of my story and trouble.

Personal story

I was diagnosed with ASD at the age of fourteen. The first few years were hard, and little progress was made. In a radical move, a therapist decided to tell me the diagnosis was wrong; to this day, I cannot decide if this was misguided. Either way, some progress was made when I stopped thinking I could not do things because of my diagnosis. Progress stopped after a few months, but the problems did not go away. When I was aged 21, I slowly grew more emotionally unstable; near the end of my bachelor's degree of science at the age of 22, I sought out therapy again. I was retested, with a clear diagnosis of ASD. I was devastated; I felt broken. However, in a few weeks time I would start searching scientific literature and some stories shared by others with a diagnosis, and finally fully appreciated the neurodiversity perspective.

I dedicated myself to learning what specific traits made me different. For instance, I often encounter severe problems with visual explanations. Often visual explanations only make sense if I translate them into text or mathematics; yet equally often I can not figure them out. Another important trait is that large groups of people are emotionally draining, and going to lectures involving larger groups destroys my concentration and retention for most of the day. Many misunderstand me when I say larger groups; usually, it is about ten people.

I finished my bachelor, and entered the master's degree with the lessons learned. I've now trained myself sufficiently that people simply reject my diagnosis - yet I know better. I have not learned "how to behave normal", but how to function. I'm no longer trying to emulate, but to communicate. And that makes a lot of difference. I still have only a few friends, by choice. I have learned discipline, I have learned how to function with my impairments. I've learned how to minimise communication issues I encounter.

And I've learned never to stop learning. Even now, I revise social situations, trying to figure out facial expressions or other non-verbal communication I had misinterpreted. Before a presentation, I practise non-verbal communication at home; I make faces into a mirror and practise gestures. By now, non-verbal communication is becoming habit, as intended.

Things to avoid.

I would like people to avoid a few things. First, don't call me high-functioning. Would you call a professional sporter "lucky to be talented"? By calling me high-functioning, you are implying that I only have a minor disability that can be easily managed. As far as my feelings are concerned, you are dismissing the effort I make in managing my disability.

Second, please refrain from saying that you have some traits of autism. Everyone does, as the NG article shows. I understand that it's not intended, but it sounds almost as if you're saying you're similar, so that it can't be all that bad.

Third, don't start about people you know that have a diagnosis, especially not that you know people that are successful. I know people with autism can be successful, and I don't need to be reminded. I don't need encouragement.

Fourth, never start me on Einstein. I adore Einstein in most respects, and his scientific work is exemplary. However, you know probably know very little about him and are very likely to repeat myths. Einstein was never diagnosed and his IQ was never measured.

Finally, do not try and explain to me how some woo you've read explains all that. Don't blame it on vaccines. Don't try to get me on a gluten-free diet. Please don't tell me to "detox". Just accept that I am different.

Things to do.

What would I like people to do? First, try not to rely on nonverbal communication. It is often hard for me to read. For instance, when you tell an anecdote it is a lot simpler to add your verdict; this leaves no room for misinterpretation.

The second is to remind me of my errors. I fear repeating errors and distantiating people by errors that I am ignorant of.

The third is to allow for my planning. I understand that people like spontaneity, but for me that is often stressful. I need to spend a lot of time alone, and sudden changes can still make me anxious.

Finally, do these things in general. Clarity in communication is becoming more and more important, as you encounter people with less social and cultural overlap. It has become clear that a large part of the population has trouble with such things; it is far simpler to make a habit of accounting for these.


We've gone over what autism is, what origins it has and some of the ways in which I've experienced it. Furthermore, I have given some suggestions on things to avoid and things to apply.

I sincerely hope that this has given you some insight. Perhaps it will make things easier for those in your environment, even if you don't know of them. After all, some of these traits are shared even by people without a diagnosis.