Il Nuovo Cimento A (1965-1970)

, Volume 62, Issue 3, pp 722–754

# Revision of the relativistic dynamics with variable rest mass and application to relativistic thermodynamics

• G. Cavalleri
• G. Salgarelli
Article

## Summary

For point bodies with variable rest mass it is shown that the definition of forcem0dU i /dτ=Fi is better than the usual d(m0Ui))/dτ=Fi and the explicit equation of motion is given bym0dUi/dτ=F i ext +(z−1U i B Ui)dm0/dτ, where the external forceF i ext has been separated from the reaction force due to the ejected mass,U i B is the four-velocity of the mass centre of the ejected mass andz=U s B U2c−2. For extended bodies it is shown that the von Laue mechanical-energy flux (in the usual «synchronous» formulation) is not plausible. Moreover a new formulation, called «asynchronous» is given. By this formulation many problems receive an immediate solution. The resultant «asynchronous» force density is always orthogonal to the four-velocity, and the equations of motion for continuous media turn out to be formally equal to the classical equations. Expanding the new equations to the first order when the pressure is a regular function of space and time, one obtains the usual «synchronous» equations. In the asynchronous formulation, all quantities relevant to extended bodies transform in a covariant way. In particular, momentum and energy turn out to have the expression proposed by Rohrlich. The above concepts are applied to relativistic thermodynamics. It is shown that, for point bodies, heat transforms as in the recent Ott formulation. For extended bodies, in the usual «synchronous» formulation, heat transforms as pointed out by von Laue. In the «asynchronous» formulation we always haveQ=γQ0. TemperatureT is considered invariant and the relation dQ=TdS valid in the rest system only.

## Keywords

Lorentz Transformation Relativistic Dynamic Extended Body Rest System Point Body
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

# Улучшение релятивистской динамики с переменной массой покоя и применение к релятивистской термодинамике

## Резюме

Показывается, что для точечных тел с переменной массой покоя определения силыm0dUi/dr=Fi является лучше, чем обычное определение d(m0Ui)/dr=Fi, и приводится точное уравнение движенияm0dUi/dr=F i ext +(z−1U i B Ui)dm0/dr где внешняя силаF i ext была отделена от силы реакции, обусловленной испущенной массой,U i B представляет четырех-скорость центра масс для испущенной массы иz=U 8 B U3c−2. Для протяженных тел показывается, что механический поток энергии фон Лауэ в обычной «синхронной» формулировке не является правдоподобным. Кроме того, приводится новая формулировка, называемая «асинхронной». С помощью этого формализма непосредственно получаются решения многих проблем. Результирующая «асинхронная» плотность силы оказывается всегда ортогональной четырех-скорости, и соответствующие уравнения движения для непрерывной среды формально совпадают с классическими уравнениями. Разлагая новое уравнение до первого порядка, когда давление является регулярной функцией пространства и времени, можно получить обычные «синхронные» уравнения. В асинхронном формализме все величины, относящиеся к протяженным телам преобразуются ковариантным образом. В частности, оказывается, что импульс и энергия имеют то же выражение, как было предложено Рорлихом. Вышеуказанные концепции применяются к релятивистской термодинамике. Показывается, что для точечных тел тепло преобразуется согласно преобразованию в недавней формулировке Отта. Для протяженных тел в обычном «синхронном» формализме тепло преобразуется, как указано по фон Лауэ. В «асинхронном» формализме всегдаQQ0. температура рассматривается инвариантной, и соотношение dQ=TdS справедливо только в системе покоя.

## Riassunto

Per corpi puntiformi con massa a riposo variabile, viene mostrato che la definizione di forzam0dUi/dτ=Fi è migliore dell’usuale d(m0Ui)/dτ=Fi e che l’equazione di moto esplicita è data dam0dUi/dτ=F i ext +(z−1U i B Ui)dm0/dτ, dove la forza esternaF i ext è stata separata dalla forza di reazione dovuta alla massa espulsa,U i B è la tetravelocità del centro di massa della massa eiettata ez=U s B U5c−2. Per corpi estesi viene mostrato che il flusso di energia meccanica di von Laue (nell’usuale formulazione «sincrona») non è plausibile. Inoltre viene data una nuova formulazione, chiamata «asincrona». Per mezzo di questa formulazione molti problemi ricevono una soluzione immediata. La densità della forza risultante «asincrona» è sempre ortogonale alla tetravelocità e le equazioni di moto per i mezzi continui risultano formalmente uguali alle equazioni classiche. Sviluppando al prim’ordine le nuove equazioni quando la pressione è una funzione regolare dello spazio e del tempo, si ottengono le usuali equazioni «sincrone». Nella formulazione asincrona tutte le grandezze relative a corpi estesi si trasformano in modo covariante. In particolare la quantità di moto e l’energia vengono ad avere l’espressione proposta da Rohrlich. I concetti di cui sopra vengono applicati alla termodinamica relativistica. Viene mostrato che per corpi puntiformi, il calore si trasforma come nella recente formulazione di Ott. Per corpi estesi e nell’usuale formulazione sincrona, il calore si trasforma come messo in evidenza da von Laue. Nella formulazione asincrona si ha senpreQ=jQ0. La temperaturaT è considerata invariante e la relazione dQ=TdS valida solo nel sistema di riposo.

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