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Reference: Einstein’s 1920 Book

Section VI (Part 1)
The Theorem of the Addition of Velocities Employed in Classical Mechanics

Please see Section VI at the link above.

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Summary

In classical mechanics, we have the theorem of addition of velocities when there is a system of reference that itself is moving relative to another system of reference. But this theorem does not always hold in reality.

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Comments

When a body of reference is moving, its internal inertia also plays a part. That is why different bodies have different uniform velocities inherently. But, in material domain, the difference in inertia is so small that it is practically imperceptible.

Therefore, in most cases in the material domain the inertia can assumed to be constant. This allows velocities of the same order of magnitude to be simply added. But when the two velocities are not of the same order of magnitude, the theorem of addition cannot be applied.

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Reference: Einstein’s 1920 Book

Section IV (Part 1)
The Galileian System of Co-ordinates

Please see Section IV at the link above.

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Summary

The Law of Inertia basically means that in the absence of external forces, a body moves without acceleration at an inherent constant velocity in a straight line.

If we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day. But this is opposed to the statement of the law of inertia, since the visible fixed stars have no motion or acceleration. We, therefore, conclude that the earth is actually rotating around an axis. Thus, the law of inertia also indicates the reference-bodies, or systems of co-ordinates, permissible in mechanics.

The “Galilean system of co-ordinates” is a system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it. The laws of the mechanics of Galilei-Newton can be regarded as valid only for a Galilean system of co-ordinates.

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Comments

The law of inertia is very specific about zero acceleration and the uniform velocity of the body in a straight line. However, it omits to mention anything about the magnitude of that uniform velocity.

The magnitude of the uniform velocity may depend on the mass of the body, because if the mass of the body is infinite, it cannot have inherent velocity.

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Reference: Einstein’s 1920 Book

Section II (Part 1)
The System of Co-ordinates

Please see Section II at the link above.

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Summary

In reality, the body of reference is the rigid body with the point with which the position of an object or an event coincides. A length is correct only when it is measured directly with the use of a rigid rod of standard length.

Distance to locations that are far from the body of reference can be determined by means of optical observations according to the rules and methods laid down by Euclidean geometry, and taking into account the properties of the propagation of light.

We may apply the Cartesian system of co-ordinates to describe a position in space. Such a system shall consists of three rigid plane surfaces perpendicular to each other and rigidly attached to the body of reference. Perpendiculars can be dropped from that position to those three plane surfaces to describe it by means of three coordinates.

The measurements are generally made by indirect means as described above. However, the physical meaning of specifications of position must always be sought.

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Comments

“Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body.” ~Einstein

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