How much time does a mass take to experience gravitational attraction due to another mass? Is it instantaneous or time delayed? Will it take some time for a person to feel the weight of a jug full of water? Sir Isaac Newton although did not mention the time in experiencing gravity in his popular Laws Of Gravitation yet it is universal that gravity is felt instantaneously without loss of any time. The gravitational laws simply state the force of attraction between two masses is proportional to their product divided by square of the distance between them.
Precession Of The Orbit Of Mercury
It was observed that planet mercury precesses by a small angle in its orbit and Newton’s laws of gravitation were not able to explain this behaviour. Different hypothesis were put forth to explain precession of the orbit but in vain.This anomalous precession of mercury was successfully explained by French astronomer Urbain Le Verrier. He theorised that force of attraction between two masses not only depends upon inverse square of distance but also upon the cube of distance between them.
It was a big set back to Newton’s inverse square law. These were reduced to the status of approximate laws. It, not only hit laws of gravitation but also fortified the theory of General Relativity as dependence of gravitational force upon the cube of distance was a mere deduction from the laws of general relativity. Dependence of gravitational force upon GR was the grand success of theory of General Relativity GR and it elated Einstein to the extent that he felt heart palpitations with the happiness.
Laplace Derivation Of Velocity Of Gravity
In modern physics, force is attributed to interaction of particles of the the field. Gravitational force is interactions of hypothetical massless particles called gravitons. It appears obvious that for exerting instantaneous attraction, the gravitons must propagate at infinite velocity.
Is it possible to have superluminal (moving with velocity higher than that of light) particles? Before Einstein’s Theory of relativity, there was no bar upon the value of maximum achievable velocity of a particle. Gravitational laws as postulated by Newton were impregnable. There was no bar upon highest achievable velocity.
That led physicists to give their hypothesis based on velocities that far exceeded that of light. First attempt to determine the speed of gravity was made by Laplace, he considered the gravitational field as radiation field whereby changes in the velocity of the attracting masses transmit some sort of waves. While examining orbital motion of a planet, he proposed that speed of gravity must be appreciably high to cancel the effect of delay in time to cause gravitational attraction as this delay was likely for interacting particles to reach the masses which were situated apart. He determined the velocity of gravity to be at least 7×10^6 times the velocity of light.
This hypothesis was also rejected as Einstein theory of Special relative was proved experimentally correct on the fateful day of solar eclipse. With that, the limitation of maximum achievable velocity by any object was fixed as that of light abbreviated as ‘c.’ Now hypothesis of infinite speed of particles gravitons constituting gravity was unacceptable and unscientific.
Theoretical Value Of Velocity Of Gravity According To GR
If we consider solutions of field equations of GR, these provide theoretically the value of velocity of gravity as that of speed of light. That means gravitons interactions propagate at velocity c. If such were the case, would it not take some time to experience gravity due to heavy object situated at an astronomical distance? To consider the effect of delay in time in experiencing gravity, we take the case of solar system where light rays reach earth in 500 seconds from sun. Let us take the hypothetical case where the sun disappears suddenly at time T. Light rays which started at time T from sun will continue falling on earth for 500 seconds more after disappearance of the sun. Let us examine the case of gravity, considering velocity of gravity as that of light, will the earth also continue experiencing gravitational attraction for 500 seconds more after the disappearance of sun?
As it is stated earlier, solutions to General Relativity provide velocity of gravity equal to that of light c. In that situation, Gravity would not cease immediately and earth orbital motion must continue for 500 seconds more. If it were assumed that the earth would continue orbiting 500 seconds more then that would mean, changes in gravitating mass would not cause immediate change in gravitational field at a distance. But this explanation is too simple to prove speed of gravity and its interaction and it requires further corroboration.
Instability Of Orbit
Take the orbital motion of Jupiter around the sun, at any time, force due to gravity felt by the sun would pertain to gravitational attraction that took place earlier by time ‘t’at Jupiter where t is time taken by gravitational interactions to reach from Jupiter to the sun. That is, the sun is experiencing gravity from the Jupiter at its present position earlier by time t. Similarly, the Jupiter will also experience gravity from the sun at its earlier position. In such a situation, the sun is attracting the Jupiter at its earlier position say ghost position. And the Jupiter is attracting the sun from its ghost position.
We know, for a stable orbit, line of action of the force between the sun and the Jupiter must be directed towards them. But in the case being discussed, there are two separate parallel forces and we know that such parallel forces yield a couple which will destabilise the system. Therefore, effect of the gravitational force must be experienced towards their present positions instantaneously irrespective of the fact that the sun and the Jupiter are situated at a distance. That warrants infinite velocity of gravity to cause its effect instantaneously but GR warrants velocity of gravity to be equal to that of light.
Concept Of Spacetime
This is a paradoxical situation which needs resolution. General Relativity lays down that mass and energy warps or curves the spacetime. Spacetime in simple words can be considered as expression of space by four variables including time. We know, a point came be expressed with three coordinates ie length, width and height or x, y and z. But the position of point may change with time t, we will have to add one more variable of time t. Spacetime of a point then can be expressed as (x,y,z,t). A ship in spacetime coordinates can be expressed at “Mumbai at time 8.00 PM on May 24, 2017.”
Coming to warping or curving of space, I submit, massive the object or energy, more warped or curved spacetime will be. In simple words, if the spacetime is highly curved, it denotes higher the value of gravity. Consider bike race in circus in a conical chamber where the bikers ply bikes all around the chamber. A biker who runs bike slow will be pulled downward and the fastest can run bike at the top rim of the chamber. If a biker stops accelerating the vehicle, he will spiral down to the floor of conical chamber. We can safe assume that the conical chamber is warped spacetime, bikers celestial bodies which orbit in warped spacetime according to their kinetic energies that depend upon their velocities. Bodies wth higher velocities will be in outer orbits and bodies with smaller velocity in inner orbits.
Instantaneous Effect Of Gravity
Having explained the basic concepts of spacetime of GR, we revert to the example where we considered the hypothetical case of disappearance of the sun and its effect on orbital motion of the earth. When the sun was there, it had the warped spacetime, when it disappeared, it is to be examined whether the warped spacetime also disappeared instantaneously with infinite speed or with finite speed of light. I submit, it is the subtle nature of the interactions that tries to annul the effect of propagation delay. That means, the characteristics nature of interactions, annul time delay in changing the warped spacetime to flat provided there is no external mass to interfere.If there were some external mass or energy, the spacetime, strictly speaking would take the form depending upon that external energy and mass.
It is worth mentioning here that these are not only the planets that experience gravitational force and orbit around the sun but the sun also experiences gravitational force and orbits the planets depending upon its mass as compared to that of planets. If total mass of the planets is very small, the star (here sun) may neither orbit nor spin.
Lorentz Invariance And Velocity Of Gravity
Before proceeding further with the subject, I state that for instantaneous disappearance of spacetime (or gravity) in the example of earth and the sun, it is necessary that there should be constant velocity between the sun and the earth. In this case the velocity is almost constant.
Gravitational field of masses which have constant speed are associated with static and continuous field that does not propagate. Lorentz invariance demands that what a moving body in a static field sees and what a moving body that emits that field sees be symmetrical. Therefore, two celestial bodies moving at constant velocity with respect to each other, each feels a force instantaneously towards the present position of the other body and therefore, there is no speed of gravity delay. In the example of the Jupiter and the sun, gravity between the two will be according to their current positions and speed of gravity delay will never be observed.
Experimental Determination Of Velocity Of Gravity
We know the gravitational force between two masses is too weak to measure, it is not possible in laboratory to measure speed of gravity. But in case of binary pulsar PSR 1913 + 16, the orbit of the binary system is slowly decreasing. The decrease in radius of the orbit is attributed to the slowing of the speed of binary stars on account of loss of energy. This loss is accrued to gravitational radiation and its lowest order is quadrupole radiation where the radiated power depends on the third time derivative of the mass quadrupole moment.
But in any field theory, radiation is related to the finite velocity of field propagation and the orbital changes due to gravitational radiation can equivalently be viewed as damping caused by the finite propagation speed. On computing the rate of damping, one finds that it depends upon the speed of gravity. If the calculations framework of GR is accepted, speed of gravity can be calculated in the basis of damping. Based on the data of this binary, the speed of gravity calculated is found equal to that of light with in 1 pc error.
I submit, speed of gravity is, in fact, speed of graviton interaction that propagates as gravitational waves and speed of gravity is, therefore, equal to speed of gravitational waves. Whenever there is disturbance in spacetime, it creates gravitational waves, obviously massive the objects, more distinctly gravitational waves can be detected.
In September 2015, pair of blackholes merged and that caused disturbance in spacetime generating gravitational waves. We have already stated that gravity propagates due to interactions of gravitons. If gravitons interaction propagates at speed of light, special relativity warrants, its rest mass must be zero. If the gravitons are found to possess mass, then they can not move with velocity of light and speed of gravity in that case would be less than that of light. And if the velocity of gravity were less than that of light, corresponding lower frequencies of gravitons would cause dispersion of light at the point of merger of the blackholes. But no such dispersion of light was observed and that also proved, the velocity of gravitational waves or gravity equals that of light with in experimental result.
In this way, experimental results and theoretical calculations concurred the velocity of gravity to be finite and equal to that of light. It is interesting to note that velocity of gravity though finite, its effect is experienced instantaneously. When a mass disappears, its spacetime also transforms to flat line instantaneously provided there is no other interfering energy or mass thanks to Lorentz invariance and subtle characteristics of spacetime. Gravitational force being always attractive, leads us to physically prove that force of attraction between two charges is always more than force of repulsion between them. It is heartening to note the characteristics of gravity that although its velocity is that of light, its effect of attraction is felt instantaneously, satisfy the works of both great physicists Sir Isaac Newton and Albert Einstein. But the fact still remains that Newton’s laws of gravitation are approximation of Einstein GR in weak gravitational field. But the gravity will continue propagating at velocity of light, 3 x 10^8 metres per second.
End
Notes:
1.References a) The Speed of Gravity – What the Experiments Say by Tom Van Flandern
b) Does Gravity Travel at the Speed of Light? by Steve Clip
2. a) Image Courtesy
Two-dimensional illustration of how mass in the Universe distorts space-time (Credit: NASA)
b) Image (Figure 1) courtesy The Speed of Gravity – What the Experiments Say by Tom Van Flandern
Writer Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
Narinder 