Curved spacetime - Knowledge

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transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock. For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.

224: 2911: 465: 2181: 1283: 2756: 2137: 310:—curved paths determined by the local geometry of spacetime—rather than being influenced directly by distant bodies. This framework led to two fundamental principles: coordinate independence, which asserts that the laws of physics are the same regardless of the coordinate system used, and the equivalence principle, which states that the effects of gravity are indistinguishable from those of acceleration in sufficiently small regions of space. These principles laid the groundwork for a deeper understanding of gravity through the geometry of spacetime, as formalized in Einstein's field equations. 2795: 1827: 327: 2128: 359:

moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.

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accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the

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length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.

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that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."

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the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10. With decades of additional lunar laser ranging data, Singh et al. (2023) reported improvement on these limits by a factor of about 100.

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More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple

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Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so

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had detected a 2 km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe (abundant in the Moon's core) and Al (abundant in its crust and mantle). If pressure did not contribute equally to spacetime curvature as does mass–energy,

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abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of

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As the astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing irregularities in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical

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The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its

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The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer

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However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 10 atm ≈ 10 Pa ≈ 10 kg·sm. This amounts to about 1% of the nuclear mass density of approximately 10kg/m (after factoring in c ≈ 9×10ms).

2173:. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as 819:

Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic

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In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small

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that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space. In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout

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General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For

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It is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit

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mounted at right angles to each other 1400 m below the Earth's surface to measure this effect. The first ten years of experience with a prototype ring laser gyroscope array, GINGERINO, established that the full scale experiment should be able to measure gravitomagnetism due to the Earth's

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In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his

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bodies or two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal

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These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated, and it is highly unlikely that any of the relevant cosmological data (e.g.

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In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform

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To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write

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curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, should not their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?

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Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the

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Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will (1976) reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.

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A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by

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over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an

382:. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be 623:(iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon 2305:. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the 1410: 4192:

Iorio, L. (February 2017). "A comment on "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. Measurement of Earth's dragging of inertial frames," by I. Ciufolini et al".

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theory of gravity will predict gravitational time dilation if it respects the principle of equivalence. This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the

2098:. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases. 619: 3035:

Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles

2771:. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant 1549: 1136:

describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to

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It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II,

2169:. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called 862: 4255: 2890:. Kreuzer found that repositioning the Teflon mass caused no differential deflection of the torsion bar, hence establishing active mass and passive mass to be equivalent to a precision of 5×10. 2820:

mission. (right) Astronomers all over the world have bounced laser light off the retroreflectors left by Apollo astronauts and Russian lunar rovers to measure precisely the Earth-Moon distance.

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seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the

1919:(red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions. 2974:

measurement, with results (as of 2016) demonstrating the effect to within 5% of its theoretical value, although there has been some disagreement on the accuracy of this result.

3076:. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along 1715: 1614: 1254:

Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859,

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to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.

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over that which could be explained by the tugs of the other planets. The ability to detect and accurately measure the minute value of this anomalous precession (only 43

1981:(orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum. 1308:

movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.

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Years before publication of the general theory in 1916, Einstein used the equivalence principle to predict the existence of gravitational redshift in the following

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Kuroda, Takami; Kotake, Kei; Takiwaki, Tomoya (2012). "Fully General Relativistic Simulations of Core-Collapse Supernovae with An Approximate Neutrino Transport".

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inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.

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so that velocities are simply additive. Fig. 5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +

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relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume

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component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all.

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To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the

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scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a

2952:, the frame-dragging effect had been confirmed to within 15% of the expected result, while the geodetic effect was confirmed to better than 0.5%. 3522:

Williams, R. K. (1995). "Extracting X rays, Ύ rays, and relativistic e–e pairs from supermassive Kerr black holes using the Penrose mechanism".

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new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.

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Williams, R. K. (2004). "Collimated escaping vortical polar e–e jets intrinsically produced by rotating black holes and Penrose processes".

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The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.

2940:(which is due to simple spacetime curvature, and is also known as de Sitter precession) to an accuracy of about 1%. The much smaller 2306: 150: 1414: 2102:

is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study

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The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by

1020:{\displaystyle \Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}-\,(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}} 4318: 3720: 3494: 3432: 3305: 3062:, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 2948:) was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes. Nevertheless, by 2297:, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve. 2252:

The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle

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states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.

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in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.

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from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:

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gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a

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The Ginger project – preliminary results. Proceedings of the MG16 Meeting on General Relativity Online; 5–10 July 2021

3299: 3212: 3184: 3152: 3116: 200: 3813: 2027:. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the 801:{\displaystyle {\frac {E'}{E}}={\frac {h\nu \,'}{h\nu }}={\frac {m}{m+{\frac {mgh}{c^{2}}}}}=1-{\frac {gh}{c^{2}}}} 2285:

Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal

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In Fig. 5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −

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calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.

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An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation,

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Everitt; et al. (2011). "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity".

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Bartlett, D. F.; Van Buren, Dave (1986). "Equivalence of active and passive gravitational mass using the moon".

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In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of

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Kreuzer, L. B. (1968). "Experimental measurement of the equivalence of active and passive gravitational mass".

3353:"Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète" 2095: 1845:

In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the

386:), there are no experiments that A and B can perform which will enable them to tell which setting they are in. 342:

In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of

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The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity

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Another effort, the Gyroscopes in General Relativity (GINGER) experiment, seeks to use three 6 m

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space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.

3398: 2266: 1619: 1215: 1100: 339:. No evidence of gravitation can be discovered following alongside the motions of a single particle. 4127:"A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model" 3448:

Thorne, Kip S. (1988). Fairbank, J. D.; Deaver, B. S. Jr.; Everitt, W. F.; Michelson, P. F. (eds.).

2055:. Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, 4372: 1883: 1852: 1846: 125: 3909: 3999: 3908:

Singh, Vishwa Vijay; Müller, Jürgen; Biskupek, Liliane; Hackmann, Eva; Lämmerzahl, Claus (2023).

2829: 2755: 2274: 2153: 262: 2925:, a satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2196:, available online.) Analogous logic can be used to demonstrate the origin of gravitomagnetism. 2145:

Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass–energy distributions with

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Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time

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Ciufolini, Ignazio; Paolozzi, Antonio Rolf Koenig; Pavlis, Erricos C.; Koenig, Rolf (2016).

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Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean

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One important conclusion to be derived from the equations is that, colloquially speaking,

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really represent tidal effects resulting from their differential attraction by mass

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There is no "action and reaction" principle dictating any necessary relationship between

2107: 4216: 4152: 4087: 3935: 3878: 3829: 3790: 3641: 3588: 3535: 4347: 4228: 4202: 4169: 4138: 4126: 4107: 4073: 3921: 3653: 3627: 3600: 3574: 3458: 3331: 3174: 2991: 2269:. It has been proposed that such gravitomagnetic forces underlie the generation of the 1180: 1160: 1140: 852:

as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance

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general relativity, but it is not possible to demonstrate any non-trivial derivations.

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Fig. 5-5 classifies the various sources of gravity in the stress–energy tensor:

1255: 180: 4111: 3462: 3050:, concludes that a force of repulsion exists between the particles in scenario 4220: 4164: 4156: 4095: 4091: 3943: 3939: 3882: 3833: 3794: 3645: 3604: 3592: 3539: 3287:

Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity

2930: 2767:(Fig. 5-9a). Two small but dense balls are suspended on a fine wire, making a 2290: 2174: 2146: 1826: 1259: 654: 3484: 3285: 3106: 2937: 2919: 2768: 2321: 834: 319: 303: 95: 1405:{\displaystyle \Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}} 4004: 3712:

The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference

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Subsequent measurements of frame dragging by laser-ranging observations of the

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Matter in motion through a gravitomagnetic field is hence subject to so-called

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Altucci, C.; Bajardi, F.; Basti, A.; Beverini, N.; Capozziello, S. (2021).

4178: 4103: 3951: 3894: 3551: 3399:"May 29, 1919: Eddington Observes Solar Eclipse to Test General Relativity" 3330:

Carroll, Sean M. (2 December 1997). "Lecture Notes on General Relativity".

2302: 2127: 299: 294:, gravity naturally arises, as opposed to being described as a fundamental 190: 4278:"Einstein right using the most sensitive Earth rotation sensors ever made" 3054:

while a force of attraction exists between the particles in scenario

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Figure 5–11. Gravity Probe B confirmed the existence of gravitomagnetism

492:(Fig. 5-3) has been constructed. (ii) Drop a particle of rest mass 2310: 1276: 447:. The equivalence principle states that these two masses are identical. 3336: 3684: 3077: 2817: 80: 46: 38: 20: 1655:

predicts bending of light around a star. In general relativity, the

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Comptes rendus hebdomadaires des séances de l'Académie des Sciences

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If pressure does not act as a gravitational source, then the ratio

2286: 2166: 363: 287: 4078: 3715:. Berlin, Germany: Max Planck Research Library. pp. 159–162. 3632: 2338:

Bondi distinguishes between different possible types of mass: (1)

2333: 2309:, the process becomes runaway and the neutron star collapses to a 2256:

This velocity-dependent gravitational effect is gravitomagnetism.

4248:"Underground ring lasers will put general relativity to the test" 3749: 2803: 2301:

example, the pressure term sets a maximum limit to the mass of a

1849:, the sources of gravity are presented on the right-hand side in 614:{\displaystyle m+{{\tfrac {1}{2}}mv^{2}}/{c^{2}}=m+{mgh}/{c^{2}}} 468:

Figure 5–3. Einstein's argument suggesting gravitational redshift

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Bondi, Hermann (1957). DeWitt, Cecile M.; Rickles, Dean (eds.).

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Figure 5-5. Contravariant components of the stress–energy tensor

1258:, in an analysis of available timed observations of transits of 3489:(New Millenium ed.). Basic Books. pp. 13–6 to 13–11. 2961: 2944:

effect (which is due to gravitomagnetism, and is also known as

4000:"Perseverance Is Paying Off for a Test of Relativity in Space" 3139:

Lorentz, H. A.; Einstein, A.; Minkowski, H.; Weyl, H. (1952).

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from the top of the tower. It falls freely with acceleration

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Figure 5–9. (A) Cavendish experiment, (B) Kreuzer experiment

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of a gravitational force acting at a long range from Earth.

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In special relativity, mass-energy is closely connected to

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that is moving in the same direction as the bottom stream.

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Figure 5–4. General relativity is a theory of curved time

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Figure 5-10. Lunar laser ranging experiment. (left) This

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Experimental test of the sources of spacetime curvature

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Hobson, M. P.; Efstathiou, G.; Lasenby, A. N. (2006).

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inertial frame. In small enough regions of spacetime,

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Two central propositions underlie general relativity.

3754:. Fullerton, CA: Light and Matter. pp. 241–258. 3179:. Princeton, New Jersey: Princeton University Press. 3108:

Spacetime Physics: Introduction to Special Relativity

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should be lower for nuclei with higher atomic number

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Mathematical theory of the geometry of space and time

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terms (green) represent isotropic pressure, and the

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and Einstein's rise to fame is well told elsewhere.

4033:"Gravity Probe B Science Results—NASA Final Report" 3483:Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). 3376:"The Hunt for Vulcan, the Planet That Wasn't There" 2872:, in which the electrostatic pressures are higher. 2516:The third law of action and reaction dictates that 1200:But general relativity is a theory of curved space 653:, since otherwise one would be able to construct a 3140: 3105:Taylor, Edwin F.; Wheeler, John Archibald (1992). 2860: 2738: 2711: 2681: 2654: 2619: 2592: 2562: 2535: 2494: 2465: 2433: 2396: 2359: 2227: 2084: 2005: 1973: 1943: 1911: 1874: 1799: 1709: 1647: 1608: 1543: 1404: 1243: 1189: 1169: 1149: 1128: 1089: 1019: 800: 613: 3864: 3235:. Laboratoire Univers et Théories. Archived from 1821: 4339: 4030: 2982:rotation to within a 0.1% level or even better. 2918:The existence of gravitomagnetism was proven by 2114:and other phenomena in the strong field regime. 3457:. W. H. Freeman and Company. pp. 573–586. 2936:Initial results confirmed the relatively large 2334:Definitions: Active, passive, and inertial mass 515:, as measured by an observer on the ground, is 3447: 3418: 3416: 3991: 3104: 263: 4031:Everitt, C. W. F.; Parkinson, B. W. (2009). 3198: 3196: 2241:, and the bottom stream has a velocity of +2 1838:, the only source of gravitational force is 3413: 3172: 2897:In 1986, Bartlett and Van Buren noted that 3901: 3350: 3230:"Experimental Tests of General Relativity" 2816:was left on the Moon by astronauts on the 2443:is the mass which reacts to acceleration. 270: 256: 4245: 4206: 4168: 4142: 4077: 3997: 3925: 3837: 3631: 3578: 3335: 3325: 3323: 3279: 3277: 3193: 2689:is dictated by the equivalence principle. 2273:(Fig. 5-8) ejected by some rotating 1421: 953: 708: 286:is the mathematical model in which, with 3704: 3702: 3564: 3521: 3275: 3273: 3271: 3269: 3267: 3265: 3263: 3261: 3259: 3257: 3173:Mook, Delo E.; Vargish, Thoma s (1987). 3100: 3098: 3096: 2909: 2754: 2179: 1825: 1281: 816:and later by Pound & Snider (1964). 463: 362: 325: 19:For broader coverage of this topic, see 4063: 3776: 3747: 3671:Wollack, Edward J. (10 December 2010). 3670: 3486:The Feynman Lectures on Physics, vol. 2 3373: 3329: 2506:discussion of the equivalence principle 2505: 4340: 3846:from the original on 28 September 2018 3673:"Cosmology: The Study of the Universe" 3396: 3320: 3283: 3202: 3168: 3166: 3164: 2280: 2184:Figure 5–7. Origin of gravitomagnetism 2047:terms (blue) represent shear stresses. 4191: 4012:from the original on 3 September 2018 3708: 3699: 3254: 3093: 1030: 151:Newton's law of universal gravitation 4246:Cartlidge, Edwin (20 January 2016). 4045:from the original on 23 October 2012 3811: 3503:from the original on 17 January 2023 3308:from the original on 17 January 2023 3205:A first course in general relativity 500:, reaching the ground with velocity 488:: (i) Assume that a tower of height 459: 4057: 3451:Near zero: New Frontiers of Physics 3161: 3046:but unaware of the large mass 2905: 13: 3998:Gugliotta, G. (16 February 2009). 3374:Worrall, Simon (4 November 2015). 3227: 2949: 2751:Pressure as a gravitational source 2117: 1776: 1754: 1732: 1629: 1520: 1498: 1476: 1386: 1318: 1225: 1110: 1001: 979: 957: 934: 866: 657:device. We therefore predict that 630:back into a particle of rest mass 116:Introduction to general relativity 14: 4389: 4258:from the original on 12 July 2017 3758:from the original on 18 June 2017 3729:from the original on 28 July 2017 3409:from the original on 2 June 2017. 3401:. This Month in Physics History. 2627:are equal is an empirical result. 2020:of momentum per unit area in the 1816:1919 Eddington eclipse expedition 367:Figure 5–2. Equivalence principle 201:Mathematics of general relativity 121:Mathematics of general relativity 4288:from the original on 10 May 2017 3979:from the original on 2 June 2018 2802: 2793: 2307:Tolman–Oppenheimer–Volkoff limit 2135: 2126: 1710:{\displaystyle (1+2GM/(c^{2}r))} 1609:{\displaystyle (1-2GM/(c^{2}r))} 1090:{\displaystyle (1-2GM/(c^{2}r))} 236: 235: 222: 45: 4300: 4270: 4239: 4195:The European Physical Journal C 4185: 4118: 4024: 3965: 3858: 3805: 3770: 3741: 3664: 3611: 3558: 3515: 3476: 3441: 3390: 3367: 3344: 3290:(Reprint ed.). Cambridge: 3014: 2970:satellites has improved on the 1648:{\displaystyle (c\Delta t)^{2}} 1251:term dwarfs the spatial terms. 1244:{\displaystyle (c\Delta t)^{2}} 1129:{\displaystyle (c\Delta t)^{2}} 848:as the gravitational constant, 313: 4225:10.1140/epjc/s10052-017-4607-1 4161:10.1140/epjc/s10052-016-3961-8 4096:10.1103/PhysRevLett.106.221101 3944:10.1103/PhysRevLett.131.021401 3397:Levine, Alaina G. (May 2016). 3221: 3029: 3004: 2369:is the mass which acts as the 2096:gravitational potential energy 2053:gravity itself creates gravity 1836:Newton's theory of gravitation 1822:Sources of spacetime curvature 1783: 1773: 1761: 1751: 1739: 1729: 1704: 1701: 1685: 1662: 1636: 1623: 1603: 1600: 1584: 1561: 1527: 1517: 1505: 1495: 1483: 1473: 1393: 1380: 1232: 1219: 1117: 1104: 1084: 1081: 1065: 1042: 1008: 998: 986: 976: 964: 954: 941: 928: 1: 3677:Universe 101: Big Bang Theory 3405:. American Physical Society. 3086: 2375:of a gravitational field; (2) 2013:are the rates of flow of the 1875:{\displaystyle T_{\mu \nu },} 1554:In Newton's gravitation, the 410:and in Newton's second law, 292:theory of general relativity 7: 4131:European Physical Journal C 3351:Le Verrier, Urbain (1859). 3203:Schutz, Bernard F. (1985). 2985: 2861:{\displaystyle m_{a}/m_{p}} 2573:On the other hand, whether 2412:a gravitational field; (3) 511:, so that its total energy 146:Introduction to gravitation 10: 4394: 3748:Crowell, Benjamin (2000). 3650:10.1088/0004-637X/755/1/11 3292:Cambridge University Press 330:Figure 5–1. Tidal effects. 18: 3887:10.1103/PhysRevLett.57.21 3818:The Astrophysical Journal 3620:The Astrophysical Journal 3567:The Astrophysical Journal 2946:Lense–Thirring precession 2267:electromagnetic induction 1265:excess rate of precession 176:Derivations of relativity 4254:. Institute of Physics. 3799:10.1103/PhysRev.169.1007 3544:10.1103/PhysRevD.51.5387 3284:Schutz, Bernard (2004). 2997: 2275:supermassive black holes 1847:Einstein field equations 1717:coefficient in front of 1616:coefficient in front of 1208:The answer is that they 1097:coefficient in front of 814:Pound & Rebka (1959) 126:Einstein field equations 4066:Physical Review Letters 3914:Physical Review Letters 3867:Physical Review Letters 2631:In general relativity, 437:should be equal to the 91:Lorentz transformations 4313:. pp. 3956–3962. 3973:"Gravity Probe B: FAQ" 3147:. Dover Publications. 2915: 2862: 2760: 2740: 2713: 2683: 2656: 2621: 2594: 2564: 2537: 2496: 2467: 2435: 2398: 2361: 2229: 2228:{\displaystyle v\ll c} 2185: 2154:gravitomagnetic fields 2086: 2085:{\displaystyle E=mgh,} 2007: 2006:{\displaystyle T^{ij}} 1975: 1974:{\displaystyle T^{i0}} 1945: 1944:{\displaystyle T^{0i}} 1913: 1912:{\displaystyle T^{00}} 1876: 1831: 1811:of the total bending. 1801: 1711: 1649: 1610: 1545: 1406: 1296: 1293:Click here to animate. 1245: 1191: 1171: 1151: 1130: 1091: 1021: 802: 615: 469: 368: 331: 4284:. Science X network. 2913: 2863: 2758: 2741: 2739:{\displaystyle m_{p}} 2714: 2712:{\displaystyle m_{a}} 2684: 2682:{\displaystyle m_{i}} 2657: 2655:{\displaystyle m_{p}} 2622: 2620:{\displaystyle m_{i}} 2595: 2593:{\displaystyle m_{p}} 2565: 2563:{\displaystyle m_{p}} 2538: 2536:{\displaystyle m_{a}} 2512:In Newtonian theory, 2497: 2495:{\displaystyle m_{g}} 2468: 2466:{\displaystyle m_{p}} 2436: 2434:{\displaystyle m_{i}} 2399: 2397:{\displaystyle m_{p}} 2362: 2360:{\displaystyle m_{a}} 2265:effects analogous to 2230: 2183: 2087: 2008: 1976: 1946: 1914: 1877: 1829: 1802: 1800:{\displaystyle \left} 1712: 1650: 1611: 1546: 1407: 1285: 1246: 1192: 1177:, and because of the 1172: 1152: 1131: 1092: 1022: 803: 616: 467: 374:equivalence principle 366: 329: 306:. Objects move along 186:Differential geometry 86:Equivalence principle 4363:Theory of relativity 3812:Will, C. M. (1976). 2883:, while bromine has 2830: 2723: 2696: 2666: 2639: 2604: 2577: 2547: 2520: 2479: 2475:gravitational mass ( 2450: 2418: 2381: 2344: 2213: 2100:Numerical relativity 2061: 1987: 1955: 1925: 1896: 1884:stress–energy tensor 1853: 1721: 1659: 1620: 1558: 1415: 1315: 1216: 1181: 1161: 1141: 1101: 1039: 863: 844:curved space. Given 677: 521: 164:Relevant mathematics 4358:Theoretical physics 4353:Concepts in physics 4217:2017EPJC...77...73I 4153:2016EPJC...76..120C 4088:2011PhRvL.106v1101E 3936:2023PhRvL.131b1401S 3879:1986PhRvL..57...21B 3830:1976ApJ...204..224W 3791:1968PhRv..169.1007K 3642:2012ApJ...755...11K 3589:2004ApJ...611..952W 3536:1995PhRvD..51.5387W 3380:National Geographic 2899:lunar laser ranging 2775:can be determined. 2281:Pressure and stress 2194:Lectures on Physics 2108:gravitational waves 33:Part of a series on 3751:General Relativity 3425:General Relativity 2992:Spacetime topology 2916: 2858: 2761: 2736: 2709: 2679: 2652: 2617: 2590: 2560: 2533: 2492: 2463: 2431: 2406:is the mass which 2394: 2357: 2293:. Taken together, 2225: 2186: 2082: 2003: 1971: 1941: 1909: 1872: 1832: 1797: 1707: 1645: 1606: 1541: 1402: 1297: 1241: 1187: 1167: 1147: 1126: 1087: 1031:Curvature of space 1017: 798: 611: 541: 486:thought experiment 470: 429:gravitational mass 369: 332: 229:Physics portal 206:Spacetime topology 181:Spacetime diagrams 109:General relativity 81:Spacetime manifold 74:Spacetime concepts 63:General relativity 58:Special relativity 25:General relativity 4378:Conceptual models 4320:978-981-12-6977-6 3722:978-3-86931-963-6 3530:(10): 5387–5427. 3524:Physical Review D 3496:978-0-465-02416-2 3434:978-0-521-82951-9 3176:Inside Relativity 2782:of a metal ball. 2570:must be the same. 2271:relativistic jets 1814:The story of the 1461: 1373: 1256:Urbain Le Verrier 1190:{\displaystyle r} 1170:{\displaystyle M} 1150:{\displaystyle G} 921: 796: 765: 762: 722: 693: 540: 460:Curvature of time 302:static Euclidean 280: 279: 139:Classical gravity 4385: 4332: 4331: 4329: 4327: 4304: 4298: 4297: 4295: 4293: 4274: 4268: 4267: 4265: 4263: 4252:physicsworld.com 4243: 4237: 4236: 4210: 4189: 4183: 4182: 4172: 4146: 4122: 4116: 4115: 4081: 4061: 4055: 4054: 4052: 4050: 4044: 4037: 4028: 4022: 4021: 4019: 4017: 3995: 3989: 3988: 3986: 3984: 3969: 3963: 3962: 3960: 3958: 3929: 3905: 3899: 3898: 3862: 3856: 3855: 3853: 3851: 3841: 3809: 3803: 3802: 3785:(5): 1007–1011. 3774: 3768: 3767: 3765: 3763: 3745: 3739: 3738: 3736: 3734: 3706: 3697: 3696: 3694: 3692: 3683:. Archived from 3668: 3662: 3661: 3635: 3615: 3609: 3608: 3582: 3580:astro-ph/0404135 3562: 3556: 3555: 3519: 3513: 3512: 3510: 3508: 3480: 3474: 3473: 3472:on 28 July 2017. 3471: 3465:. Archived from 3456: 3445: 3439: 3438: 3420: 3411: 3410: 3394: 3388: 3387: 3382:. Archived from 3371: 3365: 3364: 3348: 3342: 3341: 3339: 3327: 3318: 3317: 3315: 3313: 3281: 3252: 3251: 3249: 3247: 3242:on 18 March 2017 3241: 3234: 3225: 3219: 3218: 3200: 3191: 3190: 3170: 3159: 3158: 3146: 3136: 3130: 3129: 3127: 3125: 3102: 3081: 3071: 3045: 3033: 3022: 3018: 3012: 3008: 2973: 2969: 2965: 2951: 2931:gravitomagnetism 2928: 2924: 2906:Gravitomagnetism 2889: 2882: 2875: 2867: 2865: 2864: 2859: 2857: 2856: 2847: 2842: 2841: 2806: 2797: 2781: 2745: 2743: 2742: 2737: 2735: 2734: 2718: 2716: 2715: 2710: 2708: 2707: 2688: 2686: 2685: 2680: 2678: 2677: 2661: 2659: 2658: 2653: 2651: 2650: 2635:The equality of 2626: 2624: 2623: 2618: 2616: 2615: 2599: 2597: 2596: 2591: 2589: 2588: 2569: 2567: 2566: 2561: 2559: 2558: 2542: 2540: 2539: 2534: 2532: 2531: 2503: 2501: 2499: 2498: 2493: 2491: 2490: 2472: 2470: 2469: 2464: 2462: 2461: 2442: 2440: 2438: 2437: 2432: 2430: 2429: 2405: 2403: 2401: 2400: 2395: 2393: 2392: 2368: 2366: 2364: 2363: 2358: 2356: 2355: 2296: 2236: 2234: 2232: 2231: 2226: 2191: 2175:gravitomagnetism 2147:angular momentum 2139: 2130: 2093: 2091: 2089: 2088: 2083: 2046: 2036: 2026: 2019: 2012: 2010: 2009: 2004: 2002: 2001: 1980: 1978: 1977: 1972: 1970: 1969: 1950: 1948: 1947: 1942: 1940: 1939: 1918: 1916: 1915: 1910: 1908: 1907: 1881: 1879: 1878: 1873: 1868: 1867: 1806: 1804: 1803: 1798: 1796: 1792: 1791: 1790: 1769: 1768: 1747: 1746: 1716: 1714: 1713: 1708: 1697: 1696: 1684: 1654: 1652: 1651: 1646: 1644: 1643: 1615: 1613: 1612: 1607: 1596: 1595: 1583: 1550: 1548: 1547: 1542: 1540: 1536: 1535: 1534: 1513: 1512: 1491: 1490: 1467: 1463: 1462: 1460: 1456: 1455: 1445: 1434: 1411: 1409: 1408: 1403: 1401: 1400: 1379: 1375: 1374: 1372: 1368: 1367: 1357: 1346: 1330: 1329: 1273:tropical century 1250: 1248: 1247: 1242: 1240: 1239: 1196: 1194: 1193: 1188: 1176: 1174: 1173: 1168: 1156: 1154: 1153: 1148: 1135: 1133: 1132: 1127: 1125: 1124: 1096: 1094: 1093: 1088: 1077: 1076: 1064: 1026: 1024: 1023: 1018: 1016: 1015: 994: 993: 972: 971: 949: 948: 927: 923: 922: 920: 916: 915: 905: 894: 878: 877: 807: 805: 804: 799: 797: 795: 794: 785: 777: 766: 764: 763: 761: 760: 751: 740: 728: 723: 721: 713: 712: 699: 694: 689: 681: 669: 664: 655:perpetual motion 652: 651: 640:It must be that 636: 628: 622: 620: 618: 617: 612: 610: 609: 608: 598: 593: 573: 572: 571: 561: 556: 555: 554: 542: 533: 510: 422: 400: 284:curved spacetime 272: 265: 258: 244: 239: 238: 231: 227: 226: 196:Curved spacetime 49: 30: 29: 4393: 4392: 4388: 4387: 4386: 4384: 4383: 4382: 4373:Time in physics 4338: 4337: 4336: 4335: 4325: 4323: 4321: 4305: 4301: 4291: 4289: 4276: 4275: 4271: 4261: 4259: 4244: 4240: 4190: 4186: 4123: 4119: 4062: 4058: 4048: 4046: 4042: 4035: 4029: 4025: 4015: 4013: 3996: 3992: 3982: 3980: 3971: 3970: 3966: 3956: 3954: 3906: 3902: 3863: 3859: 3849: 3847: 3810: 3806: 3779:Physical Review 3775: 3771: 3761: 3759: 3746: 3742: 3732: 3730: 3723: 3707: 3700: 3690: 3688: 3669: 3665: 3616: 3612: 3563: 3559: 3520: 3516: 3506: 3504: 3497: 3481: 3477: 3469: 3454: 3446: 3442: 3435: 3421: 3414: 3395: 3391: 3386:on 24 May 2017. 3372: 3368: 3349: 3345: 3328: 3321: 3311: 3309: 3302: 3282: 3255: 3245: 3243: 3239: 3232: 3226: 3222: 3215: 3201: 3194: 3187: 3171: 3162: 3155: 3137: 3133: 3123: 3121: 3119: 3103: 3094: 3089: 3084: 3063: 3037: 3034: 3030: 3026: 3025: 3019: 3015: 3009: 3005: 3000: 2988: 2971: 2967: 2960: 2938:geodetic effect 2926: 2922: 2920:Gravity Probe B 2908: 2884: 2877: 2873: 2852: 2848: 2843: 2837: 2833: 2831: 2828: 2827: 2824: 2823: 2822: 2821: 2809: 2808: 2807: 2799: 2798: 2779: 2769:torsion balance 2765:Henry Cavendish 2753: 2730: 2726: 2724: 2721: 2720: 2703: 2699: 2697: 2694: 2693: 2673: 2669: 2667: 2664: 2663: 2646: 2642: 2640: 2637: 2636: 2611: 2607: 2605: 2602: 2601: 2584: 2580: 2578: 2575: 2574: 2554: 2550: 2548: 2545: 2544: 2527: 2523: 2521: 2518: 2517: 2486: 2482: 2480: 2477: 2476: 2474: 2473:is the same as 2457: 2453: 2451: 2448: 2447: 2425: 2421: 2419: 2416: 2415: 2414:inertial mass ( 2413: 2388: 2384: 2382: 2379: 2378: 2376: 2351: 2347: 2345: 2342: 2341: 2339: 2336: 2331: 2322:nucleosynthesis 2294: 2283: 2214: 2211: 2210: 2208: 2189: 2163: 2162: 2161: 2160: 2142: 2141: 2140: 2132: 2131: 2120: 2118:Energy-momentum 2062: 2059: 2058: 2056: 2038: 2028: 2021: 2014: 1994: 1990: 1988: 1985: 1984: 1962: 1958: 1956: 1953: 1952: 1932: 1928: 1926: 1923: 1922: 1903: 1899: 1897: 1894: 1893: 1860: 1856: 1854: 1851: 1850: 1824: 1786: 1782: 1764: 1760: 1742: 1738: 1728: 1724: 1722: 1719: 1718: 1692: 1688: 1680: 1660: 1657: 1656: 1639: 1635: 1621: 1618: 1617: 1591: 1587: 1579: 1559: 1556: 1555: 1530: 1526: 1508: 1504: 1486: 1482: 1472: 1468: 1451: 1447: 1446: 1435: 1433: 1426: 1422: 1416: 1413: 1412: 1396: 1392: 1363: 1359: 1358: 1347: 1345: 1338: 1334: 1325: 1321: 1316: 1313: 1312: 1235: 1231: 1217: 1214: 1213: 1182: 1179: 1178: 1162: 1159: 1158: 1142: 1139: 1138: 1120: 1116: 1102: 1099: 1098: 1072: 1068: 1060: 1040: 1037: 1036: 1033: 1011: 1007: 989: 985: 967: 963: 944: 940: 911: 907: 906: 895: 893: 886: 882: 873: 869: 864: 861: 860: 835:Newtonian limit 790: 786: 778: 776: 756: 752: 741: 739: 732: 727: 714: 707: 700: 698: 682: 680: 678: 675: 674: 662: 658: 649: 641: 634: 626: 604: 600: 599: 594: 583: 567: 563: 562: 557: 550: 546: 531: 530: 522: 519: 518: 516: 501: 462: 446: 436: 427:reason why the 417: 411: 406: 395: 389: 387: 377: 320:reference frame 316: 304:reference frame 276: 247: 234: 221: 220: 212: 211: 210: 165: 157: 156: 155: 140: 132: 131: 130: 110: 102: 101: 100: 96:Minkowski space 75: 67: 28: 17: 12: 11: 5: 4391: 4381: 4380: 4375: 4370: 4365: 4360: 4355: 4350: 4334: 4333: 4319: 4299: 4269: 4238: 4184: 4117: 4072:(22): 221101. 4056: 4023: 4005:New York Times 3990: 3964: 3900: 3857: 3839:10.1086/154164 3804: 3769: 3740: 3721: 3698: 3687:on 14 May 2011 3663: 3610: 3597:10.1086/422304 3573:(2): 952–963. 3557: 3514: 3495: 3475: 3440: 3433: 3412: 3389: 3366: 3343: 3319: 3300: 3253: 3228:Mester, John. 3220: 3213: 3192: 3185: 3160: 3153: 3131: 3117: 3091: 3090: 3088: 3085: 3083: 3082: 3027: 3024: 3023: 3013: 3002: 3001: 2999: 2996: 2995: 2994: 2987: 2984: 2942:frame-dragging 2907: 2904: 2855: 2851: 2846: 2840: 2836: 2814:retroreflector 2811: 2810: 2801: 2800: 2792: 2791: 2790: 2789: 2788: 2752: 2749: 2748: 2747: 2733: 2729: 2706: 2702: 2690: 2676: 2672: 2649: 2645: 2629: 2628: 2614: 2610: 2587: 2583: 2571: 2557: 2553: 2530: 2526: 2510: 2509: 2489: 2485: 2460: 2456: 2428: 2424: 2391: 2387: 2377:passive mass ( 2354: 2350: 2335: 2332: 2330: 2327: 2282: 2279: 2262:frame-dragging 2224: 2221: 2218: 2144: 2143: 2134: 2133: 2125: 2124: 2123: 2122: 2121: 2119: 2116: 2081: 2078: 2075: 2072: 2069: 2066: 2049: 2048: 2000: 1997: 1993: 1982: 1968: 1965: 1961: 1938: 1935: 1931: 1920: 1906: 1902: 1871: 1866: 1863: 1859: 1823: 1820: 1795: 1789: 1785: 1781: 1778: 1775: 1772: 1767: 1763: 1759: 1756: 1753: 1750: 1745: 1741: 1737: 1734: 1731: 1727: 1706: 1703: 1700: 1695: 1691: 1687: 1683: 1679: 1676: 1673: 1670: 1667: 1664: 1642: 1638: 1634: 1631: 1628: 1625: 1605: 1602: 1599: 1594: 1590: 1586: 1582: 1578: 1575: 1572: 1569: 1566: 1563: 1552: 1551: 1539: 1533: 1529: 1525: 1522: 1519: 1516: 1511: 1507: 1503: 1500: 1497: 1494: 1489: 1485: 1481: 1478: 1475: 1471: 1466: 1459: 1454: 1450: 1444: 1441: 1438: 1432: 1429: 1425: 1420: 1399: 1395: 1391: 1388: 1385: 1382: 1378: 1371: 1366: 1362: 1356: 1353: 1350: 1344: 1341: 1337: 1333: 1328: 1324: 1320: 1290:curved space. 1238: 1234: 1230: 1227: 1224: 1221: 1186: 1166: 1146: 1123: 1119: 1115: 1112: 1109: 1106: 1086: 1083: 1080: 1075: 1071: 1067: 1063: 1059: 1056: 1053: 1050: 1047: 1044: 1032: 1029: 1028: 1027: 1014: 1010: 1006: 1003: 1000: 997: 992: 988: 984: 981: 978: 975: 970: 966: 962: 959: 956: 952: 947: 943: 939: 936: 933: 930: 926: 919: 914: 910: 904: 901: 898: 892: 889: 885: 881: 876: 872: 868: 809: 808: 793: 789: 784: 781: 775: 772: 769: 759: 755: 750: 747: 744: 738: 735: 731: 726: 720: 717: 711: 706: 703: 697: 692: 688: 685: 607: 603: 597: 592: 589: 586: 582: 579: 576: 570: 566: 560: 553: 549: 545: 539: 536: 529: 526: 461: 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