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132-397: The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times. The extended form by Albert Einstein requires special relativity to also hold in free fall and requires

264-406: A c t M 0 p a s s − M 1 a c t M 1 p a s s {\displaystyle S_{0,1}={\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}-{\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}} is used to quantify differences between passive and active mass. Tests of

396-736: A c t M 0 p a s s r 2 {\displaystyle F_{0}={\frac {M_{1}^{\mathrm {act} }M_{0}^{\mathrm {pass} }}{r^{2}}}} It follows that: M 0 a c t M 0 p a s s = M 1 a c t M 1 p a s s {\displaystyle {\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}={\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}} In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference, S 0 , 1 = M 0

528-573: A c t M 1 p a s s r 2 m 1 i n e r t = M 0 a c t M 2 p a s s r 2 m 2 i n e r t {\displaystyle {\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}m_{1}^{\mathrm {inert} }}}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}m_{2}^{\mathrm {inert} }}}} Therefore: M 1 p

660-398: A c t M 1 p a s s r 2 {\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}} Likewise the force on a second object of arbitrary mass 2 due to the gravitational field of mass 0 is: F 2 = M 0 a c t M 2 p

792-430: A s s r 2 {\displaystyle F_{2}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}}}} By definition of inertial mass: F = m i n e r t a {\displaystyle F=m^{\mathrm {inert} }a} if m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} are

924-474: A s s m 1 i n e r t = M 2 p a s s m 2 i n e r t {\displaystyle {\frac {M_{1}^{\mathrm {pass} }}{m_{1}^{\mathrm {inert} }}}={\frac {M_{2}^{\mathrm {pass} }}{m_{2}^{\mathrm {inert} }}}} In other words, passive gravitational mass must be proportional to inertial mass for objects, independent of their material composition if

1056-875: A s s m i n e r t ) A − ( m p a s s m i n e r t ) B ( m p a s s m i n e r t ) A + ( m p a s s m i n e r t ) B . {\displaystyle \eta (A,B)=2{\frac {\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}-\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}{\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}+\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}}.} Values of this parameter are used to compare tests of

1188-402: A displacement R AB , Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude where G is the universal gravitational constant . The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is This

1320-448: A quantum theory of gravity such as string theory and loop quantum gravity predict violations of the weak equivalence principle because they contain many light scalar fields with long Compton wavelengths , which should generate fifth forces and variation of the fundamental constants. Heuristic arguments suggest that the magnitude of these equivalence principle violations could be in the 10 to 10 range. Currently envisioned tests of

1452-406: A bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished groove . The groove was lined with " parchment , also smooth and polished as possible". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various angles to slow the acceleration enough so that

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1584-414: A chance to test the strong equivalence principle in a strong gravitational field with high accuracy. Most alternative theories of gravity predict a change in the gravity constant over time. Studies of Big Bang nucleosynthesis , analysis of pulsars, and the lunar laser ranging data have shown that G cannot have varied by more than 10% since the creation of the universe. The best data comes from studies of

1716-534: A curved path. "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth." Newton further reasons that if an object were "projected in an horizontal direction from

1848-513: A force from a scale or the surface of a planetary body such as the Earth or the Moon . This force keeps the object from going into free fall. Weight is the opposing force in such circumstances and is thus determined by the acceleration of free fall. On the surface of the Earth, for example, an object with a mass of 50 kilograms weighs 491 newtons, which means that 491 newtons is being applied to keep

1980-399: A friend, Edmond Halley , that he had solved the problem of gravitational orbits, but had misplaced the solution in his office. After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin for "On

2112-546: A gravitational field while still being influenced by (changing) tidal acceleration. By Newton's law of universal gravitation and laws of motion, a body of mass m at distance R from the center of a sphere of mass M feels a force F → g {\displaystyle {\vec {F}}_{g}} , equivalent to an acceleration a → g {\displaystyle {\vec {a}}_{g}} , where r ^ {\displaystyle {\hat {r}}}

2244-421: A gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body, and inversely proportional to

2376-406: A hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in free -fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum , in which there is no air resistance, the hammer and

2508-466: A hydrogen maser and comparing it to one on the ground. The Global positioning system requires compensation for this redshift to give accurate position values. Time-based tests search for variation of dimensionless constants and mass ratios . For example, Webb et al. reported detection of variation (at the 10 level) of the fine-structure constant from measurements of distant quasars . Other researchers dispute these findings. The present best limits on

2640-650: A regular monthly pattern of moonquakes on Earth's Moon. Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals, and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date. Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes , where they are responsible for

2772-400: A rocket far from any gravitational field. Since the physical laws are the same, Einstein assumed the gravitational field and the acceleration were "physically equivalent". Einstein stated this hypothesis by saying he would: ...assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system . In 1911 Einstein demonstrated the power of

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2904-570: A situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force (for example, the perturbing force on the Moon ): it is the difference between the force exerted by

3036-399: A spherical body (body 1) exerted by another body (body 2). These tidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart. The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of

3168-468: A string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body? The only convincing resolution to this question is that all bodies must fall at the same rate. A later experiment was described in Galileo's Two New Sciences published in 1638. One of Galileo's fictional characters, Salviati, describes an experiment using

3300-436: A test program incorporating two new principles—the § Einstein equivalence principle , and the § Strong equivalence principle —each of which assumes the weak equivalence principle as a starting point. Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong. Some proposals also suggest finer divisions or minor alterations. The weak equivalence principle, also known as

3432-410: A uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field. In theoretical physics , a mass generation mechanism

3564-460: A vacuum, as David Scott did on the surface of the Moon during Apollo 15 . A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle , lies at the heart of the general theory of relativity . Einstein's equivalence principle states that within sufficiently small regions of spacetime, it is impossible to distinguish between

3696-412: Is 1 ∓ 2 x + 3 x 2 ∓ ⋯ {\displaystyle 1\mp 2x+3x^{2}\mp \cdots } which gives a series expansion of: The first term is the gravitational acceleration due to M at the center of the reference body m {\displaystyle m} , i.e., at the point where Δ r {\displaystyle \Delta r}

3828-487: Is 81 times more massive than the Moon, the Earth has roughly 4 times the Moon's radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface. In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from

3960-485: Is a balance scale , which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses. Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used

4092-400: Is a unit vector pointing from the body M to the body m (here, acceleration from m towards M has negative sign). Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m . With R as the distance from the center of M to the center of m , let ∆ r be the (relatively small) distance of the particle from the center of

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4224-404: Is a distance along the axis joining the centers of m and M , a → t {\displaystyle {\vec {a}}_{t}} is directed outwards from to the center of m (where ∆ r is zero). Tidal accelerations can also be calculated away from the axis connecting the bodies m and M , requiring a vector calculation. In the plane perpendicular to that axis,

4356-450: Is a theory which attempts to explain the origin of mass from the most fundamental laws of physics . To date, a number of different models have been proposed which advocate different views of the origin of mass. The problem is complicated by the fact that the notion of mass is strongly related to the gravitational interaction but a theory of the latter has not been yet reconciled with the currently popular model of particle physics , known as

4488-463: Is absolutely equivalent to any other patch of flat space elsewhere in the universe. Einstein's theory of general relativity (including the cosmological constant ) is thought to be the only theory of gravity that satisfies the strong equivalence principle. A number of alternative theories, such as Brans–Dicke theory and the Einstein-aether theory add additional fields. Some of the tests of

4620-419: Is adequate for most of classical mechanics, and sometimes remains in use in basic education, if the priority is to teach the difference between mass from weight.) This traditional "amount of matter" belief was contradicted by the fact that different atoms (and, later, different elementary particles) can have different masses, and was further contradicted by Einstein's theory of relativity (1905), which showed that

4752-427: Is greater than R . Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of m ' s own mass, we have the acceleration on the particle due to gravitational force towards M as: Pulling out the R term from the denominator gives: The Maclaurin series of 1 / ( 1 ± x ) 2 {\displaystyle 1/(1\pm x)^{2}}

4884-416: Is measured: The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively. According to Newton's second law of motion , if a body of fixed mass m is subjected to a single force F , its acceleration a is given by F / m . A body's mass also determines

5016-457: Is much more restrictive than the Einstein equivalence principle. Like the Einstein equivalence principle, the strong equivalence principle requires gravity is geometrical by nature, but in addition it forbids any extra fields, so the metric alone determines all of the effects of gravity. If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it

5148-413: Is small compared to R , the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration a → t , axial {\displaystyle {\vec {a}}_{t,{\text{axial}}}} for the distances ∆ r considered, along the axis joining the centers of m and M : When calculated in this way for the case where ∆ r

5280-404: Is taken to follow from empirical consistency, later became known as "weak equivalence". A version of the equivalence principle consistent with special relativity was introduced by Albert Einstein in 1907, when he observed that identical physical laws are observed in two systems, one subject to a constant gravitational field causing acceleration and the other subject to constant acceleration, like

5412-412: Is the kilogram (kg). In physics , mass is not the same as weight , even though mass is often determined by measuring the object's weight using a spring scale , rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass

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5544-408: Is the acceleration due to Earth's gravitational field , (expressed as the acceleration experienced by a free-falling object). For other situations, such as when objects are subjected to mechanical accelerations from forces other than the resistance of a planetary surface, the weight force is proportional to the mass of an object multiplied by the total acceleration away from free fall, which is called

5676-430: Is the basis by which masses are determined by weighing . In simple spring scales , for example, the force F is proportional to the displacement of the spring beneath the weighing pan, as per Hooke's law , and the scales are calibrated to take g into account, allowing the mass M to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a balance measures relative weight, giving

5808-427: Is the gravitational mass ( standard gravitational parameter ) of the body causing gravitational fields, and R is the radial coordinate (the distance between the centers of the two bodies). By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from

5940-710: Is the property that (along with gravity) determines the strength of this force. In the Standard Model of physics, the mass of elementary particles is believed to be a result of their coupling with the Higgs boson in what is known as the Brout–Englert–Higgs mechanism . There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it

6072-410: Is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units,

6204-404: Is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at

6336-445: Is zero. This term does not affect the observed acceleration of particles on the surface of m because with respect to M , m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆ r

6468-467: The Cavendish experiment , did not occur until 1797, over a hundred years later. Henry Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures. Given two objects A and B, of masses M A and M B , separated by

6600-408: The Earth's magnetic field . For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by vector subtraction of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term tidal force

6732-543: The Solar System . On 25 August 1609, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were

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6864-550: The Standard Model . The concept of amount is very old and predates recorded history . The concept of "weight" would incorporate "amount" and acquire a double meaning that was not clearly recognized as such. What we now know as mass was until the time of Newton called “weight.” ... A goldsmith believed that an ounce of gold was a quantity of gold. ... But the ancients believed that a beam balance also measured “heaviness” which they recognized through their muscular senses. ... Mass and its associated downward force were believed to be

6996-405: The carob seed ( carat or siliqua ) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds , then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of

7128-495: The elementary charge . Non-SI units accepted for use with SI units include: Outside the SI system, other units of mass include: In physical science , one may distinguish conceptually between at least seven different aspects of mass , or seven physical notions that involve the concept of mass . Every experiment to date has shown these seven values to be proportional , and in some cases equal, and this proportionality gives rise to

7260-452: The ephemeris of Mars, based on three successive NASA missions, Mars Global Surveyor , Mars Odyssey , and Mars Reconnaissance Orbiter . Gravitational mass Mass is an intrinsic property of a body . It was traditionally believed to be related to the quantity of matter in a body, until the discovery of the atom and particle physics . It was found that different atoms and different elementary particles , theoretically with

7392-442: The melting point of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of the grave in 1793, the platinum Kilogramme des Archives in 1799, and the platinum–iridium International Prototype of

7524-416: The proper acceleration . Through such mechanisms, objects in elevators, vehicles, centrifuges, and the like, may experience weight forces many times those caused by resistance to the effects of gravity on objects, resulting from planetary surfaces. In such cases, the generalized equation for weight W of an object is related to its mass m by the equation W = – ma , where a is the proper acceleration of

7656-537: The speed of light (called "clock anisotropy tests") and new forms of the Michelson-Morley experiment . The anisotropy measures less than one part in 10. Testing local positional invariance divides in to tests in space and in time. Space-based tests use measurements of the gravitational redshift , the classic is the Pound–Rebka experiment in the 1960s. The most precise measurement was done in 1976 by flying

7788-453: The torsion balance pendulum, in 1889. As of 2008 , no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10 . More precise experimental efforts are still being carried out. The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance , must be absent or at least negligible. For example, if

7920-458: The " spaghettification " of infalling matter. Tidal forces create the oceanic tide of Earth 's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun . Tidal forces are also responsible for tidal locking , tidal acceleration , and tidal heating. Tides may also induce seismicity . By generating conducting fluids within the interior of the Earth, tidal forces also affect

8052-444: The "Galilean equivalence principle" or the " weak equivalence principle " has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses m and M , respectively. If the only force acting on the object comes from a gravitational field g , the force on the object is: Given this force, the acceleration of the object can be determined by Newton's second law: Putting these together,

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8184-400: The 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporated a priori in the equivalence principle of general relativity . The International System of Units (SI) unit of mass is the kilogram (kg). The kilogram is 1000 grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at

8316-492: The Earth, is directly proportional to the diameter of the Earth and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible. Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to

8448-432: The Einstein equivalence principle must be "metric theories", meaning that trajectories of freely falling bodies are geodesics of symmetric metric. Around 1960 Leonard I. Schiff conjectured that any complete and consistent theory of gravity that embodies the weak equivalence principle implies the Einstein equivalence principle; the conjecture can't be proven but has several plausibility arguments in its favor. Nonetheless,

8580-488: The Equivalence Principle and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy. With the first successful production of antimatter, in particular anti-hydrogen, a new approach to test the weak equivalence principle has been proposed. Experiments to compare the gravitational behavior of matter and antimatter are currently being developed. Proposals that may lead to

8712-675: The Kilogram (IPK) in 1889. However, the mass of the IPK and its national copies have been found to drift over time. The re-definition of the kilogram and several other units came into effect on 20 May 2019, following a final vote by the CGPM in November 2018. The new definition uses only invariant quantities of nature: the speed of light , the caesium hyperfine frequency , the Planck constant and

8844-424: The Moon's pull results in a larger difference in force between the near and far sides of Earth, which is what creates the bigger tidal bulge. Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference. The Earth

8976-428: The abstract concept of mass. There are a number of ways mass can be measured or operationally defined : In everyday usage, mass and " weight " are often used interchangeably. For instance, a person's weight may be stated as 75 kg. In a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in

9108-401: The amount of mass being accelerated. Newton, just 50 years after Galileo, investigated whether gravitational and inertial mass might be different concepts. He compared the periods of pendulums composed of different materials and found them to be identical. From this, he inferred that gravitational and inertial mass are the same thing. The form of this assertion, where the equivalence principle

9240-409: The body of mass m . For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M . If the body of mass m is itself a sphere of radius ∆ r , then the new particle considered may be located on its surface, at a distance ( R ± ∆r ) from the centre of the sphere of mass M , and ∆r may be taken as positive where the particle's distance from M

9372-401: The center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.) Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be freefalling in a straight line under the influence of

9504-405: The classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact. Albert Einstein developed his general theory of relativity starting with the assumption that the inertial and passive gravitational masses are the same. This is known as the equivalence principle . The particular equivalence often referred to as

9636-457: The degree to which it generates and is affected by a gravitational field . If a first body of mass m A is placed at a distance r (center of mass to center of mass) from a second body of mass m B , each body is subject to an attractive force F g = Gm A m B / r , where G = 6.67 × 10  N⋅kg ⋅m is the "universal gravitational constant ". This is sometimes referred to as gravitational mass. Repeated experiments since

9768-470: The double of the distance between the two bodies. Hooke urged Newton, who was a pioneer in the development of calculus , to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told

9900-434: The elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows: Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: Galileo had shown that objects in free fall under

10032-400: The equivalence principle by using it to predict that clocks run at different rates in a gravitational potential , and light rays bend in a gravitational field. He connected the equivalence principle to his earlier principle of special relativity: This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as

10164-448: The equivalence principle use names for the different ways mass appears in physical formulae. In nonrelativistic physics three kinds of mass can be distinguished: By definition of active and passive gravitational mass, the force on M 1 {\displaystyle M_{1}} due to the gravitational field of M 0 {\displaystyle M_{0}} is: F 1 = M 0

10296-458: The equivalence principle. A similar parameter can be used to compare passive and active mass. By Newton's third law of motion : F 1 = M 0 a c t M 1 p a s s r 2 {\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}} must be equal and opposite to F 0 = M 1

10428-497: The exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun. In fact, by unit conversion it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass. Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory, all objects produce gravitational fields and it

10560-410: The feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has

10692-404: The first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611, he had obtained remarkably accurate estimates for their periods. Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not

10824-402: The first paragraph of Principia , Newton defined quantity of matter as “density and bulk conjunctly”, and mass as quantity of matter. The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to

10956-436: The first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo, but

11088-404: The freely falling bodies to be massive gravitating objects as well as test particles. Thus this is a version of the equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments . It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force . It

11220-572: The gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book A Treatise of the System of the World . According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows

11352-483: The gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io . Stresses caused by tidal forces also cause

11484-421: The gravitational acceleration is given by: This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". In addition, the constant K can be taken as 1 by defining our units appropriately. The first experiments demonstrating

11616-547: The gravitational constant ( G ) depending on nearby sources of gravity or on motion, or 3) searching for a variation of Newton's gravitational constant over the life of the universe Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the Nordtvedt effect . This effect has been sensitively tested by the Lunar Laser Ranging Experiment . Up to

11748-512: The gravitational field eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies. What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that: Here local means that experimental setup must be small compared to variations in the gravitational field, called tidal forces . The test experiment must be small enough so that its gravitational potential does not alter

11880-417: The inertial mass on the left side and gravitational mass on the right side are numerically equal and independent of the material composing the masses. The equivalence principle is the hypothesis that this numerical equality of inertial and gravitational mass is a consequence of their fundamental identity. The equivalence principle can be considered an extension of the principle of relativity, the principle that

12012-580: The influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass. However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime. According to K. M. Browne: "Kepler formed a [distinct] concept of mass ('amount of matter' ( copia materiae )), but called it 'weight' as did everyone at that time." Finally, in 1686, Newton gave this distinct concept its own name. In

12144-401: The laws of physics are invariant under uniform motion. An observer in a windowless room cannot distinguish between being on the surface of the Earth and being in a spaceship in deep space accelerating at 1 g and the laws of physics are unable to distinguish these cases. By experimenting with the acceleration of different materials, Galileo determined that gravitation is independent of

12276-478: The left on the graph, meaning closer to the attracting body. For example, even though the Sun has a stronger overall gravitational pull on Earth, the Moon creates a larger tidal bulge because the Moon is closer. This difference is due to the way gravity weakens with distance: the Moon's closer proximity creates a steeper decline in its gravitational pull as you move across Earth (compared to the Sun's very gradual decline from its vast distance). This steeper gradient in

12408-418: The limit of one part in 10 there is no Nordtvedt effect. A tight bound on the effect of nearby gravitational fields on the strong equivalence principle comes from modeling the orbits of binary stars and comparing the results to pulsar timing data. In 2014, astronomers discovered a stellar triple system containing a millisecond pulsar PSR J0337+1715 and two white dwarfs orbiting it. The system provided them

12540-528: The lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about 1.1 × 10   g , while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about 0.52 × 10   g , where g is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon. The solar tidal acceleration at

12672-418: The measurable mass of an object increases when energy is added to it (for example, by increasing its temperature or forcing it near an object that electrically repels it.) This motivates a search for a different definition of mass that is more accurate than the traditional definition of "the amount of matter in an object". Tidal forces In celestial mechanics , the expression tidal force can refer to

12804-558: The motion of bodies in an orbit"). Halley presented Newton's findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three-book set, entitled Philosophiæ Naturalis Principia Mathematica (English: Mathematical Principles of Natural Philosophy ). The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and

12936-403: The nearby gravitational field. No matter how strong the gravitational field, objects in free fall are weightless , though they still have mass. The force known as "weight" is proportional to mass and acceleration in all situations where the mass is accelerated away from free fall. For example, when a body is at rest in a gravitational field (rather than in free fall), it must be accelerated by

13068-509: The object caused by all influences other than gravity. (Again, if gravity is the only influence, such as occurs when an object falls freely, its weight will be zero). Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics , Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but

13200-413: The object for one another. These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate. The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force. The tidal force acting on an astronomical body, such as

13332-430: The object from going into free fall. By contrast, on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 newtons, because only 81.5 newtons is required to keep this object from going into a free fall on the moon. Restated in mathematical terms, on the surface of the Earth, the weight W of an object is related to its mass m by W = mg , where g = 9.80665 m/s

13464-412: The orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three-millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered. Newton's cannonball was a thought experiment used to bridge

13596-426: The other body. Larger objects distort into an ovoid , and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon. When a body rotates while subject to tidal forces, internal friction results in

13728-521: The other hand, would provide a major guidepost towards unification. In addition to the tests of the weak equivalence principle, the Einstein equivalence principle requires testing the local Lorentz invariance and local positional invariance conditions. Testing local Lorentz invariance amounts to testing special relativity, a theory with vast number of existing tests. Nevertheless, attempts to look for quantum gravity require even more precise tests. The modern tests include looking for directional variations in

13860-409: The planets orbit the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of the ellipse . Kepler discovered that the square of the orbital period of each planet is directly proportional to the cube of the semi-major axis of its orbit, or equivalently, that the ratio of these two values is constant for all planets in

13992-464: The relative gravitation mass of each object. Mass was traditionally believed to be a measure of the quantity of matter in a physical body, equal to the "amount of matter" in an object. For example, Barre´ de Saint-Venant argued in 1851 that every object contains a number of "points" (basically, interchangeable elementary particles), and that mass is proportional to the number of points the object contains. (In practice, this "amount of matter" definition

14124-408: The result. The two additional constraints added to the weak principle to get the Einstein form − (1) the independence of the outcome on relative velocity (local Lorentz invariance ) and (2) independence of "where" known as (local positional invariance) − have far reaching consequences. With these constraints alone Einstein was able to predict the gravitational redshift . Theories of gravity that obey

14256-554: The results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass. In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by

14388-471: The same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body's inertia , meaning the resistance to acceleration (change of velocity ) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The SI base unit of mass

14520-503: The same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: In 1600 AD, Johannes Kepler sought employment with Tycho Brahe , who had some of the most precise astronomical data available. Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how

14652-615: The same distance r {\displaystyle r} from m 0 {\displaystyle m_{0}} then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same). a 1 = F 1 m 1 i n e r t = a 2 = F 2 m 2 i n e r t {\displaystyle a_{1}={\frac {F_{1}}{m_{1}^{\mathrm {inert} }}}=a_{2}={\frac {F_{2}}{m_{2}^{\mathrm {inert} }}}} Hence: M 0

14784-407: The same thing. Humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection: where W is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio : An early use of this relationship

14916-421: The sound of them hitting a wooden plank. Isaac Newton measured the period of pendulums made with different materials as an alternative test giving the first precision measurements. Loránd Eötvös 's approach in 1908 used a very sensitive torsion balance to give precision approaching 1 in a billion. Modern experiments have improved this by another factor of a million. A popular exposition of this measurement

15048-408: The square of the distance ( Y = 1/ X ), while the slope ( Y ′ = −2/ X ) is inversely proportional to the cube of the distance. The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to

15180-441: The square of the distance to the body's center. For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field. Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it should be theoretically possible to determine

15312-501: The strength of the Earth's gravitational field at different places, the distinction becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in kilograms ) refers to an intrinsic property of an object, whereas "weight" (measured in newtons ) measures an object's resistance to deviating from its current course of free fall , which can be influenced by

15444-426: The third body on the second and the force exerted by the third body on the first. Tidal forces have also been shown to be fundamentally related to gravitational waves . When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on

15576-500: The third on 6 April 1686–87. The Royal Society published Newton's entire collection at their own expense in May 1686–87. Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these: where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, μ

15708-556: The tidal acceleration is directed inwards (towards the center where ∆ r is zero), and its magnitude is 1 2 | a → t , axial | {\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|} in linear approximation as in Figure 2. The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example,

15840-492: The top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected." In contrast to earlier theories (e.g. celestial spheres ) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced universal gravitational mass : every object has gravitational mass, and therefore, every object generates

15972-403: The two principles are tested with very different kinds of experiments. The Einstein equivalence principle has been criticized as imprecise, because there is no universally accepted way to distinguish gravitational from non-gravitational experiments (see for instance Hadley and Durand). The strong equivalence principle applies the same constraints as the Einstein equivalence principle, but allows

16104-460: The universality of free fall or the Galilean equivalence principle can be stated in many ways. The strong equivalence principle, a generalization of the weak equivalence principle, includes astronomic bodies with gravitational self-binding energy. Instead, the weak equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Either way: Uniformity of

16236-459: The universality of free-fall were—according to scientific 'folklore'—conducted by Galileo obtained by dropping objects from the Leaning Tower of Pisa . This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless inclined planes to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös , using

16368-417: The usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course. Soon after completing work on his theory of gravity (known as general relativity ) and then also in later years, Einstein recalled the importance of the equivalence principle to his work: The breakthrough came suddenly one day. I

16500-499: The variation of the fundamental constants have mainly been set by studying the naturally occurring Oklo natural nuclear fission reactor , where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants. The strong equivalence principle can be tested by 1) finding orbital variations in massive bodies (Sun-Earth-Moon), 2) variations in

16632-443: The weak equivalence principle are approaching a degree of sensitivity such that non-discovery of a violation would be just as profound a result as discovery of a violation. Non-discovery of equivalence principle violation in this range would suggest that gravity is so fundamentally different from other forces as to require a major reevaluation of current attempts to unify gravity with the other forces of nature. A positive detection, on

16764-487: The weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping different objects and verifying that they land at the same time. Historically this was the first approach—though probably not by Galileo's Leaning Tower of Pisa experiment but instead earlier by Simon Stevin, who dropped lead balls of different masses off the Delft churchtower and listened for

16896-419: The weak equivalence principle is obeyed. The dimensionless Eötvös -parameter or Eötvös ratio η ( A , B ) {\displaystyle \eta (A,B)} is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B". η ( A , B ) = 2 ( m p

17028-461: The weak equivalence to be valid everywhere. This form was a critical input for the development of the theory of general relativity . The strong form requires Einstein's form to work for stellar objects. Highly precise experimental tests of the principle limit possible deviations from equivalence to be very small. In classical mechanics, Newton's equation of motion in a gravitational field , written out in full, is: Careful experiments have shown that

17160-535: The weight. Robert Hooke had published his concept of gravitational forces in 1674, stating that all celestial bodies have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center. In correspondence with Isaac Newton from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to

17292-551: Was done on the Moon by David Scott in 1971. He dropped a falcon feather and a hammer at the same time, showing on video that they landed at the same time. Experiments are still being performed at the University of Washington which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards dark matter in the Galactic Center . Future satellite experiments – Satellite Test of

17424-521: Was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity. Einstein's development of general relativity necessitated some means of empirically discriminating the theory from other theories of gravity compatible with special relativity . Accordingly, Robert Dicke developed

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