The general relativity theory of Einstein is built on the equivalence principle, a postulate that says that all bodies fall in the same way when suffering from gravity.

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<sub> [image credits: <a href="https://s14.postimg.cc/7ssb601pt/miscrosco.jpg">CNES</a>]</sub> </center> 
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In other words, if one takes a gravitational field (of a planet for instance) and puts some objects in it, **they will all fall with the same acceleration regardless of their nature**.

Even if this principle may sound logical, physicists since the early days (*e.g.* [**Galileo**](https://en.wikipedia.org/wiki/Galileo_Galilei)) are testing it, trying to unravel deviations. 

And observing a deviation, even at the tiniest level, will have a deep impact on [**general relativity**](https://en.wikipedia.org/wiki/General_relativity) as this would mess up its foundations.

I recall that general relativity is the theory of gravitation proposed by [**Albert Einstein**](https://en.wikipedia.org/wiki/Albert_Einstein) at the beginning of the 20<sup>th</sup> century. It describes gravity as a **geometrical property of spacetime** and how the latter is **curved by the presence of its constituents**.

General relativity is today in agreement with all experimental data, all its predictions have been verified experimentally and it is at the basis of applications like [**the GPS**](https://en.wikipedia.org/wiki/Global_Positioning_System).
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## SOME NEWTONIAN MECHANICS ##

[**Newtonian physics**](https://en.wikipedia.org/wiki/Classical_mechanics) explains that the motion of a system, that we take as a random object to make things simpler, is connected to the sum of the forces that are applied to it. 

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<sub> [image credits: <a href="https://en.wikipedia.org/wiki/Isaac_Newton">Wikipedia</a>]</sub> </center> 
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This is the famous [**second law of motion of Newton**](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton's_second_law), that states that **the sum of the forces that are applied to the object equals the product of its mass and its acceleration**.

**F** = m **a**

In the context of the free fall of objects in a gravitational field, the force has to be seen as [**the gravitational force**](https://en.wikipedia.org/wiki/Gravity#Newton's_theory_of_gravitation) felt by the object, which **is proportional to the mass of the object and to the** [**gravitational acceleration**](https://en.wikipedia.org/wiki/Gravitational_acceleration),

**F** = m **g**

In the case of our good old planet Earth, the gravitational acceleration consists in an acceleration of about 10 m/s<sup>2</sup> pointing towards the center of the planet, [**regardless of where we are on the globe**](http://nymag.com/strategist/article/best-desk-toy-snow-globe.html). But this changes depending on where we are in the cosmos.
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## GRAVITATIONAL AND INERTIAL MASSES ##

If we take the two equations above, we can equate them, so that one ends up with **g**=**a** after simplifying the masses. This means that **all bodies fall in the same way in a gravitational field**. 

However, in order to get there, we have assumed that the masses in the two above equations are the same. And this is a strong assumption!

These two masses are what are known as the [**inertial mass**](https://en.wikipedia.org/wiki/Mass#Inertial_mass) and the [**gravitational mass**](https://www.thefreedictionary.com/gravitational+mass). 

**The inertial mass that tells us to which extent an object will resist to a modification of its motion**, which can be achieved by applying some forces on it. It is the mass in the first equation in the above section.

In contrast, **the gravitational mass tells us about the sensitivity of an object to gravity**. This is the mass in the second equation in the above section.

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<sub> [image credits: <a href="https://en.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment">Saffron Blaze</a> (CC BY-SA 3.0)</a>]</sub> </center> 
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Whilst *a priori* there is nothing that predicts these two masses to be equal, there are found to be equal. Mass is mass, after all :)

This is what is called the [**equivalence principle**](https://en.wikipedia.org/wiki/Equivalence_principle).

The first who thought about an experimental test of the equivalence principle was [**Galileo**](https://en.wikipedia.org/wiki/Galileo_Galilei). He wanted to verify that **the gravitational acceleration was independent of the mass being accelerated**.

In practice, he imagined that one could let objects go from the top of the Pisa tower to verify that they were falling in the same way.
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## THE MICROSCOPE EXPERIMENT ##

[**In December last year**](https://arxiv.org/abs/1712.01176), the test of the equivalence principle moved to the next level. 

Physicists wanted to reproduce the experimental apparatus of Galileo, but **in space, at 700 km from the planet**. This gives the chance to be **agnostic from any local surface modification of the gravitational field**, and to be thus more precise. 

This was achieved thanks to the [**MICROSCOPE mission**](https://microscope.cnes.fr/en/MICROSCOPE/index.htm), a satellite launched exactly 2 years ago.

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<sub> [image credits: <a href="https://microscope.cnes.fr/en/first-gravitational-results-microscope">CNES</a>]</sub> </center> 
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The principle of the experiment is easy. **Two different masses** (one in titanium and one in platinum) **are maintained exactly in the same orbit with the help of forces that are applied to them**. 

Those forces are measured, and from their relative comparison, one concludes whether the equivalence principle holds. 

As in space, one can control any perturbation at a very good level, **the test of the equivalence principle is done at an unprecedented level of precision**. 

It was found that **the relative difference in the [free-fall](https://en.wikipedia.org/wiki/Free_fall) accelerations of the titanium and platinum alloys where of at most 10<sup>-15</sup>**. This means that the inertial and gravitational masses are the same, up to the first 14 digits!

The mission is now about to end, and the analysis of new data will allow us to improve the precision with an extra factor of 10. But there will be no way to be more precise because of the motion of the satellite itself. We are reaching the limit!
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## TAKE-HOME MESSAGE ##

The equivalence principle states that the gravitational mass (how gravity is felt) and inertial mass (how a motion can be modified through the application of forces) are the same. Tests of this principle are made since the 17<sup>th</sup> century (at the time at which it was coined).

Recently, the MICROSCOPE experiment published it first results (see [**this article**](https://arxiv.org/abs/1712.01176)) where the equivalence principle was tested in space. By studying the free fall of two different objects at 700km from Earth, it was shown that they were falling in the same way at the 10<sup>-15</sup> level of precision.

This is very good for general relativity that strongly relies on the equivalence principle. However, we all know that general relativity cannot be the ultimate theory (as it does not work well with quantum mechanics). Therefore, it is worthy to pursue the tests to the next level of precision. Hopefully, more results in a close future!

<sub> As usual, something irrelevant is hidden in this post. </sub>

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