Curvature: computations with illustrative formulas

Curvature, yes. Earth’s curvature is a topic about which a number of calculations are available: you just need surfing the net and search for everything you want. A great number of images, posts, and videos show full evidence there is no curvature. Notwithstanding, I want to discuss the topic all the same. In fact, it touches our senses in a very sensible way. I’m going to deal first with computations and then with some example.

In the following calculation, I would like to show what is the fall you can expect for a given distance on the surface of the earth. As a start, I have to say that I’m going to show the whole reckoning but, if you are not interested in it, you can simply check the final formula. In fact, it is very handy to prove the Earth is a plane surface.


1)The observer is on the sea level

S is an arc and it is the given distance on the Earth’s surface. R is the Earth radius (6378 Km), X is the curvature fall at the s distance when the observer is on the point O on the Earth’s surface.

We can reason as Eratosthenes did: s is a fraction of the circumference of the Earth (40000km), and, proportionally, α is a fraction of the total angle of 360°. We can thus write this proportion:

From the above proportion you find:

Then, from trigonometry, you’ll obtain:

which is the final formula that calculates the curvature.

2) The observer is standing high above the sea level

Let’s consider now the case in which the observer is not on the surface but at a height Y.

In the above image s is the total distance considered that creates a α angle at the center of the Earth. Y is the height of the observer from Earth’s surface. X is the curvature fall at the distance s and it is our unknown datum.

We can write exactly as before:

And then:


In conclusion, this is the exact formula to calculate the curvature, when considering the observer at a certain height.

Curvature: examples

A friend of mine passed me this picture taken at Mentone in France. As you will notice, from there it is possible to see the Corse Island.

Let’s try to use the formula just obtained above. I want to check if the earth’s curvature could allow me to look that far.

As you can notice, the total distance from coast to coast, at the closest point, is about 175km. But, if you prefer considering the highest point inside the Corse island, you’ll have to measure the distance from Mentone to the top of Mount Cinto, at 2706m on the sea level. In the above picture, I actually consider that point. And, in that case, you’ll have a distance of about 195 Km.

Let’s suppose my friend has taken the picture at an altitude of about 10 meters above the sea level. For instance, you could check,  a site that permits (by clicking on a point on Google map) to determine the altitude of a spot on the sea level. As you can note, in this case, you have an altitude of 5,31 meters. Anyway, I’ll fix 10 meters, as an approximation.

Illustrative formulas

So, we can refer to the following formula:

Considering a coast 175 km far, we’ll obtain:


This is the fall of the curvature within a distance of 175 Km. Thus, it should be clear that, for such a distance, it wouldn’t be possible to see anything more than the top of the mountain. But there’s something more to highlight. Pay attention, the distance of this mountain from Mentone is a little farther. Let’s try to make the calculation for a distance of 195 Km and check if it is possible, from Mentone, to see at list the top of the mountain.

For a distance of 195 km, we can obtain:


Unfortunately, the mountain is only 2706m tall. Maybe, the observer could go a little higher and see at least the summit.

So, when we consider the observer as standing at an altitude  of 7 meters (that is more realistic) we obtain:


Anyway, in this image, we can see a good deal of land. How is that possible?

It is just possible because the Earth is flat.

Curvature: a few objections

1) About refraction

Objection 1: refraction of the air is what makes the phenomenon possible, even if the Earth is a globe.

Answer: Ok, let’s analyze what is the atmospheric refraction and then try to understand something more.

Refraction is the deviation of a ray of light while passing through the atmosphere and it is due to the variation of density of the air with the height. The air is, in fact, denser at sea level and rarefies going higher. Refraction makes celestial bodies to appear higher than they are.

However, when we are on the ground level and observe an object on the Earth surface, we are on the same layer of the atmosphere, with very small density variation. It is different from the case when we observe a celestial body, very high in the sky.

So, the situation we are considering can’t be a refraction phenomenon, since the light doesn’t pass through different density layers of the atmosphere.

2) About Einstein’s General Relativity

Objection 2: The light is bent by the gravitational mass of the Earth according to Einstein’s general relativity theory. Hence the Corse is visible due to the fact that the light bends along the curvature of the Earth.

Answer: Einstein postulated that the gravitational field produces a deformation of the space-time and, according to his equivalence principle, any physical entity, regardless of mass, equally accelerates in a gravitational field. Einstein made a calculation applied to a light beam grazing the sun and obtained:

That is a very small angle: 0,00024 degrees. If we want to do the same reckoning for the Earth we’ll obtain an angle:

β=0.000287 arcsecond = 8E-8 degree

which, at a distance of 195 km, produces a variation in the height of 0,27 millimeters. It is clear that this phenomenon has nothing to do with relativistic considerations.

The bending of the light

However, in relation to the Einstein’s bending of light, we have to remark that not all the scientists agree with his theory. Einstein proved its rightness by measuring the bending of the light of a star during an eclipse, in 1919. The experiment is still remembered as a complete success. Though, many and many times experiments give good results only in the imagination and theoretical data can differ a lot from reality.

Here are some words to explain this theory:

“Einstein’s law of gravitation contains nothing about force. It describes the behavior of objects in a gravitational field – the planets, for example – not in terms of ‘attraction’ but simply in terms of the paths they follow. To Einstein, gravitation is simply part of inertia; the movement of the stars and the planets arise from their inherent inertia; and the courses they follow are determined by the metric properties of space – or, more properly speaking, the metric properties of the space continuum” (Lincoln Barnett, The Universe and dr. Einstein, London, June 1949, page 72).

Einstein’s experiment

Einstein concluded his theory by saying that the light bends in the curved space-time near a big mass such that of the sun. He suggested that this could be verified by an experiment. It could be made measuring the track of the light of stars near the sun during an eclipse. That is the only moment when the sun and the stars can be seen together in the sky.

The photo of the star, twinkling from behind the sun, was taken during the eclipse. It had to be compared with pictures taken in other moments. That is to say when only the stars were visible in the sky, and, the deflection of light had to be evident through a different position of them. The light of the stars should bend inward because of the space-time curvature generated by the big mass of the sun. The theoretical value for that experiment was a bending of 1,75 arc/seconds.

The Eclipse expected was that of 29 may 1929 and it was visible in the equatorial regions. A measurement was taken in western Africa, at Principe Island, in the city of Roca Sundy. Even if the weather was not favorable, pictures were taken, and, the result was a bending of 1,64 second arc, very near to the expected result.

Some objections against Einstein’s experiment

I want here to report a consideration made by the Captain of the Indonesian Navy, Gatot Soedarto. In his book “True, General Relativity is wrong”, he made the following notation: “The proving method for hypothesis, as suggested by Einstein as the theory founder, should not be able to be carried out, considering the fact that in scientific exposure in astronomy, the instant observation applies. It means all calculations to determine the ‘true position’ and ‘the apparent position’ of a certain star at the sky is only applicable at a certain time and at a certain place on which such observation is performed. The observation on a star conducted twice from the places with different geographical positions will result in the different height and azimuth of the star…Therefore the test should not be able to be performed.”

Due also to refraction, the star, seen at different times of the day, will be seen in different places in the celestial sphere, making this experiment a complete error, even considering the very small angle measured, much smaller in respect with the refraction angles.

Soedarto continues: “In astronomy, the light deviation is something very common and not caused by gravity field of massive object, but it occurs due to the light refraction.”

And he also states that, in that year, another expedition in North East of Brazil returned a measured bending value of light of 0.93 seconds of arc, no more so close to the theoretical value. This big difference has been ignored and this second experiment has been forgotten as it had never existed.

Curvature: the conclusion

These data show that often what is generally accepted as correct and has been spread as a very well experimented datum and a valid scientific principle, maybe is not the truth but, for some reasons, the system wants to spread a false truth to hide the reality. A great and increasing number of scientists raise doubts against Einstein’s theory that however continues to be publicized as the scientific truth.

In conclusion, it is not possible a relativistic effect on light.

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