195 Longitudinal and transverse
mass
Subject
In the treatment of special relativity, sometimes a longitudinal and a transverse mass is introduced. This is to express that the inertia of a body is different (greater) in the direction of movement than in the direction transverse to it.
Deficiencies
The need to introduce two new mass concepts arises, if one insists, that mass should be a measure for the inertia. In fact, the inertia of a body moving at relativistic speed is greater in the direction of motion than transversely.
Two remarks in this regard:
1. Irrespective of whether the mass does us the favour of measuring inertia or not, we want to ask ourselves the question what to under-stand by inertia in the context of a process of movement. It is rea-sonable to define an inertia T as follows:
T := F/a (1)
and this always, i.e. not only in the case of classical movements where the force is proportional to the acceleration, i.e. when
T = m .
We bring equation (1) into another form. With a = dv/dt and F = dp/dt we get
T := dp/dv
The inertia defined in this way tells us how much momentum dp must be supplied to a body so that its velocity changes by dv.
Since we know the relativistic relation between p and v, we can easily calculate the inertia. For a change of momentum in forward direction we find
and for the transverse direction
Let us first have a look at the inertia in forward direction. It is neither identical with the rest mass nor with the relativistic mass
This is easy to see when looking at the p(v) relation, Fig. 1. T is giv-en by the slope of the curve, i.e. the differgiv-ential quotigiv-ent dp/dv, see the red tangent to the curve. The relativistic mass, however, is equal to the slope of the green straight line. Only at the beginning, in “classical approximation”, the slope dp/dv is equal to p/v, and thus equal to the rest mass, see the blue tangent.
Fig. 1. The inertia of a body is given by the slope of the function p(v). It depends on the velocity.
Now to the transverse inertia: It is that of a body which does not move in the transverse direction. However, this does not mean that it is described by the rest mass, since the mass of the body has in-creased due to the high longitudinal velocity.
In short: inertia is a quantity, which has a greater value in a given, well-defined direction than in the orthogonal direction or in other words: it is a tensor.
2. Should we conclude that there exists another tensorial mass be-sides the rest mass and the relativistic mass? This is not a good question. A physical quantity exists, if we introduce it, if we define it. Let us try to ask the question in a better way: Should we introduce a tensorial inertial mass in addition to the rest mass and the relativistic mass? A cautious answer would be: We should do so, if it is useful, if it is worthwhile. And is it worthwhile? The answer to this question is probably rather: No.
But isn’t it a pity about the beautiful interpretation of mass as a uni-versal measure of inertia?
A pity perhaps – but why should mass be better off than other physi-cal quantities? Let us remember:
• When we construct or invent a new theory, we are happy if the variables it contains measure simple properties known to us from our everyday experience. Most of the time, however, this does not quite work. Think of force, for example, or heat.
• The inertia behaves similar to some electrical quantities. The re-sistance characterizes an object: a resistor. If somebody says that the resistor has a resistance of 10 kΩ, then one is informed. However, this is only possible if the current is proportional to the voltage. But what if it is not? How do we characterize for example a semiconductor diode? In this case it is not enough to give one number. One has to give the U-I characteristic curve. The same applies to the capacitance. And we are in the same situation with the inertia. Inertia cannot be described by a single number; one needs a characteristic curve, Fig. 1.
Origin
The concepts of longitudinal and transverse mass were introduced by Lorentz in 1899 and they were also calculated by Einstein in 1905 using his theory of relativity. Since then, they have been haun-ting physics, although they have no apparent use.
Disposal
With the rest mass and the relativistic mass there are enough mass-es, not to mention the possibility to introduce consequently a longi-tudinal and a transverse energy. Nothing is missing if the longitudi-nal and transverse masses are ignored. The fact that a body has dif-ferent inertia in the forward and transverse directions can be ac-commodated in an exercise, but introducing two new terms would be a bit too much of a good thing.
All that is to be understood in this context is contained in the dia-gram of Fig. 1. It becomes even clearer, if one does not, as usual, plot momentum versus velocity, but rather velocity versus momen-tum, Fig. 2, because as independent variable one chooses, if possi-ble, that quantity, on whose values one has the most direct influence – and that is not velocity, but momentum. We push the accelerator pedal so that the engine pumps momentum from the earth into the car, and see on the speedometer what consequence this has, i.e. what velocity results from it.
Fig. 2. The function v(p) tells us everything about the inertial behavior of a body.
But what happens then to the nice rule that mass is a measure of inertia? Well, we have to relativize that a bit: It measures inertia only as long as the speed is not too high. Only for v << c, inertia is an in-trinsic property of a body, and does not depend on its state.
Tl(v )= m0 1–v 2 c2 ⎛ ⎝⎜ ⎞ ⎠⎟ 3 2 Tt(v )= m0 1–v 2 c2 m(v )= m0 1–v 2 c2
