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# Gravity and Velocity

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## Abstract

The Doppler effect is that when the Austrian physicist Christian John Doppler accidentally observed a train passing by him in 1842, he found that when the train was getting closer and closer to him, the sound of the siren became more and more, bigger and sharper, and as it gradually moved away from him, the whistle sound became smaller and softer. Therefore, by studying this kind of phenomenon, he summarized the important natural law of "the phenomenon that makes the observer feel the frequency of the wave changes obviously when the observer and the wave source move relative to each other.". The essence of the Doppler effect is that the relative velocity between objects will affect the effect between them. Not only does the wave have this characteristic, but the interaction between objects has a similar effect, which we can call the general Doppler effect or the chasing effect. This article will deduce the relationship between gravity and velocity based on Newton's gravity equation, and demonstrate the Doppler effect of gravity.
Gravity and Velocity
tony1807559167@gmail.com
Tony Yuan
Beihang University, China
January 5, 2022
Abstract: The Doppler eﬀect is that when the Austrian physicist Christian John
Doppler accidentally observed a train passing by him in 1842, he found that when the
train was getting closer and closer to him, the sound of the siren became more and more,
bigger and sharper, and as it gradually moved away from him, the whistle sound became
smaller and softer. Therefore, by studying this kind of phenomenon, he summarized the
important natural law of ”the phenomenon that makes the observer feel the frequency of
the wave changes obviously when the observer and the wave source move relative to each
other.”. The essence of the Doppler eﬀect is that the relative velocity between objects
will aﬀect the eﬀect between them. Not only does the wave have this characteristic,
but the interaction between objects has a similar eﬀect, which we can call the general
Doppler eﬀect or the chasing eﬀect. This article will deduce the relationship between
gravity and velocity based on Newton’s gravity equation, and demonstrate the Doppler
eﬀect of gravity.
Keywords: Newtonian gravity; Doppler eﬀect; gravitational constant;
1 Introduction
Newtonian gravity is a force that acts at a distance. No matter how fast an object
travels, gravity acts upon the object instantaneously. Gravity is only related to the
mass and distance of the object, equal to G0M m
r2, of which the universal gravitational
constant G0= 6.67259 ×1011 Nm2/kg2.G0is measured when two objects are rel-
atively stationary. This can be regarded as a static gravitational constant. Newtonian
gravity states that the speed of the gravitational ﬁeld on an object is inﬁnite, there-
fore, whether two objects are relatively stationary or moving, both can be considered
unchanged. Therefore, under the premise that the gravitational speed is inﬁnite, if the
Doppler eﬀect  exists, it can be ignored. General Relativity (GR), its view on the speed
of the gravitational ﬁeld is diﬀerent from that of Newton and Laplace. GR states that
the speed of the gravitational ﬁeld is equal to the speed of light. At present, there is still
some controversy over the speed of the gravitational ﬁeld . For the convenience of
1
research, we use Xto represent the velocity of the gravitational ﬁeld. No matter whether
Xis inﬁnite or equal to c, the velocity of ordinary objects is very small compared to X.
Next, so we must ﬁrst ﬁgure out what is the relationship between gravity and velocity,
and whether this relationship is the Doppler eﬀect.
2 Derivation of the Relationship between Gravity and Velocity based
on Newton’s Gravity Equation
In a very short time slice dt, we can assume that mis stationary and the gravity received
is constant. We can then accumulate the impulse generated by the gravity on each
time slice and ﬁnd the average relating to the entire time period to obtain eﬀective
constant gravitation and determine the relationship between the equivalent gravitation
and velocity.
Figure 1: Gravity model
As shown in Figure 1, there are two objects with masses Mand m, the distance
between them is r,mhas a moving velocity relative to M, the speed is v, in the absence
of other external forces, due to the gravitational force between Mand m,vwill change,
but this change is very small and can be ignored, so in a short period of time vcan
be regarded as stable, and the direction of the velocity is depicted by the straight line
connecting them. F(t) = G0M m
(r+vt)2represents the gravity on mat time t. The Newtonian
equation of gravity is used here. In any small time dt,mcan be regarded as stationary.
An accumulation of the impulse dp is obtained by multiplying the gravity and time in
these small time slices. Then, the sum of the gravitational impulse received by mwithin
a certain period can be obtained. Supposing that the gravitational impulse obtained by
mis pafter time Thas passed, the gravity is integrated into the time domain:
p=
T
Z
0
F(t)×dt =
T
Z
0
G0Mm
(r+vt)2×dt =G0M m
r2×
T
Z
0
1
(1 + vt/r)2×dt, (1)
p=G0Mm
r2×r/v
1+vt/r |T
0,
p=G0Mm
r2×T
1+vT /r .
2
For an object mwith a speed of v, the accumulated impulse pduring time Tcan be
expressed by an equivalent constant force multiplied by time T. For the convenience of
description, we use F(v) to express this equivalent force.
F(v) = p/T =G0Mm
r2/(1 + vT
r).(2)
Let us continue to think about the diﬀerence in the average gravitational force re-
ceived by two objects at diﬀerent speeds during time T? Assuming that two objects
have diﬀerent speeds, v1=v0δv,v2=v0+δv, their average gravity:
F(v1) = F(v0δv) = G0M m
r2/1 + (v0δv)T
r.
F(v2) = F(v0+δv) = G0M m
r2/1 + (v0+δv)T
r.
F(v1)F(v2) = G0M m
r2r
r+(v0δv)Tr
r+(v0+δv)T.
F(v1)F(v2) = G0M m
r22rδvT
(r+v0T)2(δvT )2.
When Tis inﬁnitesimal close to 0, F(v1)F(v2) = G0M m
r22δvT
r.
Assume K=T
r, δv =v1v2, so the following formula is obtained:
F(v1)F(v2) = G0M m
r2(Kδv),(3)
we can see that there is a linear relationship between average gravity and velocity, when
Tis inﬁnitesimal close to 0, the average gravity is the instantaneous gravity.
Based on the above analysis, we know that there is a linear relationship between
gravity and velocity, and this relationship is the general Doppler eﬀect. According to
this relationship between the object and the gravitational ﬁeld, the boundary conditions
of the general Doppler eﬀect are introduced:
1. When an object’s velocity relative to the source of gravity is 0, it is Newtonian
gravity.
2. When an object’s velocity relative to the gravitational ﬁeld is 0, the gravitational
force no longer acts on the object.
As shown in the Figure 2, according to the general Doppler eﬀect (chase eﬀect), using
boundary conditions F(0) = G0Mm
r2and F(X) = 0, it can be easily calculated:
F(v) = F(0) + v×F(X)F(0)
X=F(0) ×Xv
X=G0Mm
r2×Xv
X, the gravitational
equation with a parameter of vis sorted out as follows:
F(v) = G0M m
r2×f(v), f(v) = Xv
X.(4)
If it is necessary to preserve the form of Newton’s gravity equation, we may write it
as follows:
F(v) = G(v)×Mm
r2, G(v) = G0×Xv
X.(5)
3
That is, the gravitational constant becomes a function of v,G(v). Thus, we may
understand that when the gravitational ﬁeld has a diﬀerent velocity relative to m, the
gravitational constant is also diﬀerent.
Figure 2: Linear relationship between gravity and velocity
If we need to consider the scene where the velocity direction of the object mis
inconsistent with the direction of the gravitational ﬁeld, we deﬁne vras the component
of the velocity in the direction of the gravitational ﬁeld, and then obtain the general
equation:
F(vr) = G0M m
r2×Xvr
X.(6)
This equation shows that when an object has a velocity component in the direction
of the gravitational ﬁeld, that is, there is a motion eﬀect in the same direction between
the gravitational ﬁeld and the object, and the gravitational force received is reduced;
when the object has a velocity component opposite to the direction of the gravitational
ﬁeld, that is, there is a eﬀect of opposing movement, and the gravitational force received
increases.
3 Conclusion
We use Newton’s equation of universal gravitation and mathematical calculus and limit
calculations to prove that gravity has a Doppler eﬀect. In this way, under the same dis-
tance, the velocity of the object is diﬀerent, the gravitational force will also be diﬀerent.
If according to Newton and Laplace’s point of view, the speed of gravity is very huge,
then the general Doppler eﬀect of gravity is negligible; if according to GR’s point of
view, the speed of gravity is equal to the speed of light, for objects with very high speed,
such as the Sun and planets, the general Doppler eﬀect of gravity cannot be ignored.
4
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 Fritz Rohrlich (25 August 1989). From Paradox to Reality: Our Basic Concepts of
the Physical World. Cambridge University Press. pp. 28–. ISBN 978-0-521-37605-1.
 Newtonian Gravity http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node35.html
 ”2018 CODATA Value: Newtonian constant of gravitation”. The NIST Reference on
Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
 ”Doppler Shift”. astro.ucla.edu.
 The Detection of Gravitational Waves using LIGO, B. Barish Archived 2016-03-03
at the Wayback Machine
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Neutron Star–Black Hole Coalescences”. The Astrophysical Journal Letters. 915 (1).
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 Barry C. Barish, Caltech. The Detection of Gravitational Waves. Video from CERN