LECTURE III GRAVITATION AND THE GRAVITATIONAL FLELD A. THE IDENTITY OF GRAVITATIONAL, CENTRIFUGAL AND INERTIAL MASS As seen in the preceding lecture, the conception of the ether as the carrier of radiation had to be abandoned as incompatible with the theory of relativity; the conception of action at a distance is repugnant to our reasoning, and its place is taken by the conception of the field of force, or, more correctly, the energy field. The energy field is a storage of energy in space, character- ized by the property of exerting a force on any body susceptible to this energy — that is, a magnetic field on a magnetizable body, a gravitational field on a gravitational mass, etc. Light, or, in general, radiation, is an electromagnetic wave — ^that is, an alternation or periodic variation of the electromagnetic field^ — and the difference between the alternating fields of our transmission lines, the electro- magnetic waves of our radio stations, the waves of visible light and the X-rays are merely those due to the differences of frequency or wave length. The energy field at any point of space is determined by two constants, the intensity and the direction, and the force exerted by the field on a susceptible body is proportional to the field intensity and is in the direction of the energy field. Thus the force exerted by the magnetic field on a magnetic material is: F = HP (1) 46 GRAVITATION AND THE GRAVITATIONAL FIELD 47 where H is the magnetic field intensity and P the magnetic mass, the same quantity which in the days of action at a distance was called the magnetic pole strength, and which is related to the magnetic flux $ by: $ = AtP. The force exerted by an electric field on an electrified body is: F=KQ, (2) where K is the dielectric field intensity and Q the electric mass or electric quantity, also called electrostatic charge, measured in coulombs. The force exerted by a gravitational field is : F = gN, (3) where g is the gravitational field intensity and N the sus- ceptibility of the body to a gravitational field, or the gravitational mass of the body- — often simply called the mass. The force exerted by a centrifugal field is F = CR, (4) where C is the centrifugal field intensity and R the centri- fugal mass. The force F acting on a body exerts an acceleration a and thus produces a motion, a velocity v. The acceleration produced by the force is proportional to the force and inversely proportional to the resistance of the body against being set in motion — that is, the ability of the body in taking up kinetic energy, in other words, the inertial mass M — which thus is defined by the equation: W = Mvy2, (5) where W is the kinetic energy taken up by the mass M to give it the velocity v. The acceleration produced by the force thus is : a = F/M, (6) 48 RELATIVITY AND SPACE and, substituting in (6) the expressions of the force, in equations (1) to (4), we get: Force: Acceleration: Magnetic field F = HP a = HP/M ] Electric field F = KQ a = KQ/M Gravitational field F = gN a = gN/M Centrifugal field F = CR a = CR/M (7) The acceleration given to a body in a field thus is pro- portional to the field intensity and to the energy mass (magnetic mass, electric mass, etc.) and inversely propor- tional to the inertial mass of the body. That is, if I bring into the same magnetic field H two bodies of the same mass M — that is, two bodies which would require the same kinetic energy W to be given the same velocity v — and if these two bodies have two different magnetic masses, as a piece of cast iron and a piece of wrought iron, then the accelerations a will be different and the bodies will acquire different velocities. Or, inversely, two bodies of the same magnetic mass P in the same field H, but of different inertial masses M, would have different accelerations and so would be set in motion with different velocities. In the same manner in an electric field two identical bodies^ — ^that is, bodies of the same mass M- — having differ- ent electric charges Q would have different accelerations and so acquire different velocities. Experience, however, shows that in a gravitational field as well as in a centrifugal field all bodies have the same acceleration a and thus acquire the same velocity. That means that the gravitational mass N is the same as the inertial mass M, and the centrifugal mass R is the same as the inertial mass M, in the equations (7). This is a startling conclusion, as the gravitational mass A^ is the susceptibility of the body to the action of the gravi- tational force, just as the magnetic mass P is the suscepti- bihty to the action of the magnetic force and as such has GRAVITATION AND THE GRAVITATIONAL FIELD 49 nothing to do with the mass M, which is the storage capac- ity of the body for kinetic energy. There is no more reason why the inertial mass M should be the same as the gravitational mass N than there is that M should be the same as the magnetic mass P or the electric charge Q. This strange identity of two inherently uncorrelated quantities, the inertial mass and the gravitational mass, usually is not realized, but in writing the equation of the kinetic energy we write: W = Mvy2, and when expressing Newton's law of gravitational force we write: F = M.M^/P. That is, we use the same symbol M, call it mass, and never realize that there is no reason apparent why the ''mass" in Newton's law of gravitation should be the same thing as the ''mass" in the equation of kinetic energy. If thus the gravitational mass equals the inertial mass, there must be some relation between the gravitational force and the inertia of moving bodies. B. CENTRIFUGAL FORCE AS A MANIFESTATION OF INERTIA With regard to the centrifugal force we know this, and know that the centrifugal force is not a real force, but is merely the manifestation of the inertia in a rotating system, and it is natural, then, that the "mass," which enters into the equation of centrifugal force should be the same as the inertial mass M: R = M. Fig. 10. Let (in Fig. 10) 5 be a body revolving around a point 0. The fundamental law of physics is the law of inertia. 50 RELATIVITY AND SPACE "A body keeps the same state as long as there is no cause to change its state. That is, it remains at rest or continues the same kind of motion — that is, motion with the same velocity in the same direction — until some cause changes it, and such cause we call a 'force.'" This is really not merely a law of physics, but it is the fundamental law of logic. It is the law of cause and effect: "Any effect must have a cause, and without cause there can be no effect." This is axiomatic and is the fundamental conception of all knowledge, because all knowledge con- sists in finding the cause of some effect or the effect of some cause, and therefore must presuppose that every effect has some cause, and inversely. Applying this law of inertial motion to our revolving body in Fig. 10: A point P of the periphery, moving with the velocity of rotation v in tangential direction, would then continue to move in the same direction, PQ, and thereby move away from the center 0, first slowly, then more rapidly^ — -that is, move in the manner in which a radial or centrifugal acceleration a acts on P, or an apparent force F = Ma — by equation (6) — and this we call the centrifugal force. It is obvious, then, that all bodies would show the same centrifugal acceleration, as all would tend to move in the same manner in the same direction, unless restrained by a force (as the force of cohesion of the revolving body), and that for this reason ''centrifugal mass" is identical with the inertial mass. C. THE LAW OF GRAVITATION The identity of the gravitational mass with the inertial mass then leads to the suspicion that the gravitational force also is not a real force, but merely a manifestation of inertia, and Einstein has shown that the laws of the gravi- tational force are identical with the laws of inertial motion in an accelerated system. GRAVITATION AND THE GRAVITATIONAL FIELD 51 Let (in Fig. 11) C be a railway car standing still on a straight level track and B a billiard table in the train. I put a billiard ball A on the table, and it stands still until I push it; then it moves in a straight line at constant speed — that is, obeys the laws of inertial motion. To me in the car and to the observer on the track the behavior of the billiard ball is the same. Suppose now (in Fig. 12) the car moves at constant veloc- ity y on a straight level track. If I, being in the car, put a ball on the billiard table, it stands still until I push it, then _Q £ ^ R c -^v T T Fig. 11. Fig. 12. it moves at constant speed in a straight line, just as it did in Fig. 11, when the car stood still. Thus from the inside of the car I cannot distinguish whether the car is moving or standing still. The observer from the track sees the billiard ball standing still relative to the billiard table, but moving at constant speed together with the billiard table and the train. To me, in the car, and relative to the car, it seems to stand still; and when I push it its motion with reference to the observer on the track is the resultant of the motion of the ball relative to the train and the motion of the train, but still the motion is inertial motion to me in the moving car as well as to the observer on the track. Now the car reaches a 10 per cent grade and runs up this grade at constant speed y, as shown in Fig. 13. If I now put a billiard ball on the table, it does not remain at rest, but starts moving toward the back of the train, first slowly, then with increasing velocity, and if I push the ball across the table, it does not move in a straight line, but curves backward toward the end of the train; that is, it has an acceleration due to the force acting on it. This force is a component of the force of gravity, due to the grade p = 10 52 RELATIVITY AND SPACE per cent, and the acceleration thus is a = 0.1^ = 2.2 miles per hour per second. Suppose now, however, that the train is again running on a straight level track, not at constant speed, but at an increasing speed; that is, it is accelerating, at an acceleration a = 2.2 miles per hour per second, as shown in Fig. 14. If now I put a ball on the table, it does not remain at rest, but starts and moves backward with increasing velocity. If I push the billiard ball across the table, it does not run at <^-<-r) B Fig. 14. constant speed in a straight line, but curves backward, just as it did in Fig. 13 on a 10 per cent up grade at constant train speed. That is, an acceleration a, and thus appar- ently a force, acts on it. In short, from the inside of the car I cannot distinguish whether the train is climbing a grade at constant speed or accelerating on a level, because the effect of the acceleration of the train is identical with the effect of the force of gravitation as it acts on the billiard ball on an up grade. To the observer from the track, however, there is a difference between the motion of the billiard ball in Fig. 13 and in Fig. 14. In Fig. 14, the observer from the track does not see any force acting on the billiard ball, and the ball moves at constant speed in a straight line, in inertia! motion; but the train, and thus the billiard table, acceler- ating, slide forward under the ball, so that relative to the billiard table the ball seems to move with accelerated motion. Thus there is no acceleration and no force acting on the biUiard ball for the observer on the track. In Fig. 13, however, on an up grade, the observer on the track notices the same acceleration acting on the billiard ball as I do in the train — that is, the same force acts on the ball GRAVITATION AND THE GRAVITATIONAL FIELD 53 relative to the observer from the track as acts on it relative to me in the train. This is obvious, since the observer on the track is in the same gravitational field. It thus follows that the laws of inertial motion with regard to an accelerated system are the same as the laws of motion in a gravitational field. The former, however, are derived without any physical theory, merely as a mathe- matical transformation of the laws of inertial motion to an accelerated system. The law of gravitation thus appears here as such a mathematical transformation to an acceler- ated system and has been derived in this manner by Einstein. For all velocities which are small compared with the velocity of light Einstein's law of gravitation and Newton's law give the same results, and a difference appears only when the velocity of the moving bodies approaches in magnitude the velocity of light, as is the case, for instance, with ionic motions. Thus the gravitational field is identical with the mani- festation of inertia in an accelerated system, and the law of gravitation appears as the mathematical transformation of the equation of inertial motion in fieldless space to the equation of the same motion relative to an accelerated system. The gravitational field thus is identical with an accelerated system and can be replaced by it, and, inversely, motion relative to an accelerated system can be replaced by a gravitational field. This does not mean that any gravitational field (like that of the earth) can be replaced by some physically possible form of acceleration, but merely that the equations of motion are the same and that any limited gravitational field^ — ^for instance, that in a room — can be replaced by an acceleration in the direction of the lines of gravitational force. The force of gravitation thus has followed the centrifugal force in being resolved into a manifestation of inertial motion, and an analogy thus exists between the centrifugal 54 RELATIVITY AND SPACE force as the (apparent) effect of the acceleration in a rotating system and the gravitational force as the effect of a rectilinear accelerating system. If, however, gravitational force is a manifestation of inertial motion, it becomes obvious that the gravitational mass is identical with the inertial mass, just as the centrif- ugal mass is identical with the inertial mass. D. CENTRIFUGAL FORCE AND GRAVITATIONAL FORCE It is interesting to follow this analogy somewhat further. Suppose we consider a revolving body hke the earth. The equation of the centrifugal force is: F^ ^ Ma; (8) that is, mass times acceleration. Or, if v is the tangential velocity and I the radius of the revolving body: Fc = Mvyi (9) The gravitational force is in opposite direction to the centrifugal force. Thus, if we give the one the positive sign, we would give the other the negative sign. As the effect of the centrifugal force is to increase, that of the gravitational force is to decrease the distance between the acting bodies, the negative sign may be given to the latter. The gravitational force, then, is: F^ = -Ma, (10) that is, mass times acceleration. We may give the gravitational force the same form by introducing a fictitious velocity v, as acceleration is of the dimension velocity square divided by length, writing: Fo = -Mvyi (11) or: F, = M{-vyi) = M {jvY/l (12) = Mvy I GRAVITATION AND THE GRAVITATIONAL FIELD 55 where Vo = jv is an imaginary velocity, and F^ then has the same equation as F^. Gravitation thus appears as the centrifugal force of an imaginary velocity. An "imaginary velocity" on first sight appears unreason- able and meaningless. But it is no more so than, for instance, a negative force, as in equation (10). A "nega- tive force" inherently has no meaning, but we give it a meaning as representing a force in opposite direction. But just as the negative sign represents the opposite direction, so the imaginary sign represents the quadrature direction. That is, an imaginary velocity is a velocity at right angles, just as a negative velocity would be a velocity in opposite direction. As the velocity v in the equation of the centrifugal force is the tangential velocity, the imaginary velocity ^o = jv in the equation of the gravitational force is the velocity at right angles to the tangential velocity — that is, it is the radial velocity- — and the gravitational force then appears as the centrifugal force of radial motion, and inversely. Thus here, by mere mathematical formalism, we get the same relation between centrifugal and gravitational force as the effect of inertia in the acceleration due to tangential and radial motion. E. DEFLECTION OF LIGHT IN THE GRAVITATIONAL FIELD It is interesting to note the difference regarding the mass M between Newton's law of gravitation and Einstein's law. In Newton's law of gravitation the mass cancels. That is, the force of gravitation is : F = cjM, where g is the gravitational field intensity, M the mass. The acceleration produced by the force F is : a = F/M 56 RELATIVITY AND SPACE and substituting for F gives : a = gM/M = g. The gravitational acceleration thus equals the gravitational field intensity. In Einstein's law of gravitation the mass M does not enter at all. Einstein's law of gravitation is the mathematical trans- formation of the motion of A, in Fig. 14, to an accelerated system. But whether A is a material body like a billiard ball, or a mathematical point, or an immaterial thing like a beam of light, has no effect on the mathematical equations. Neither does it make any difference whether the body A belongs to the accelerated system or enters it from the outside. For instance: Let (in Fig. 15) i? be a railway car moving at constant speed y on a straight level track, as seen from the top. I, C s, B \. -co \ Vf Fig. 16. leaves the car, at the point B of the track, it is greater and is v^. Then the angle which the bullet makes relative to the car is tan coi = Vilv^ at the entrance of the bullet at A and is tan C02 = Vijv^ (thus being greater) when the bullet leaves the car. Thus while the bullet moves in a straight line AB relative to the track, relative to the car it curves backward, starts in the direction ABi, as if it came from Oi, but leaves in the direction B^Oi, as if coming from Oi. Hence it curves back just like the billiard ball in Fig. 14; that is, an acceleration and a gravitational field act on it. Now, there obviously is no difference in the apparent motion of OABC relative to the train whether it is a 58 RELATIVITY AND SPACE material body like a rifle bullet, or a mathematical point, or a beam of light. In other words: A gravitational field acts on a beam of light in the same manner as it acts on a material body, and a beam of light in a gravitational field is deflected and curves. A curvature necessarily means that the velocity at the inside of the curve is less than at the outside. Thus in a gravitational field the velocity of light is not constant, nor does the light move in a straight line, but it is slowed down and deflected. At first this seems to contradict our premise, that the velocity of light is constant and the same everywhere. However, this applied only to the velocity of light in empty space. In a material body the velocity of light is less. This follows from the phenomena of refraction. (In the same manner the velocity of propagation of electrical energy in a conductor is slowed down.) We get now a more com- plete understanding of the meaning of ''empty space"; that is, empty space means a space free from matter and free from energy — matterless and fieldless space — ^and the law is: ''The velocity of light in empty space, that is, in space containing no matter and no field of force, is constant, and its path a straight line, with regard to any system of reference." Assume thus (in Fig. 16) a beam of light, of velocity c, traversing the car, while the velocity of the car increases from Vy to V2. The light then enters the car at the angle, relative to the car, of tan wi = Vi/c, and leaves the car at the angle of tan w^ = v^/c. It is deflected by the accelera- tion of the car^ — that is, by the (apparent) gravitational field existing in the car — by the angle: CO = CO 2 — COi. (13) As V]_ and Vi are small compared with c, we can substitute the angle co for tan co; that is coi = Vi/C C02 = V^/C (14) GRAVITATION AND THE GRAVITATIONAL FIELD 59 Thus: CO = (^2 — Vi)/C. (15) If, now, t is the time required by the beam of light to traverse the car, and g the acceleration of the car, it is: Vi - Vi = gt; (16) that is, the increase of velocity is acceleration times time. Substituting (16) into (15) gives: co= gt/c (17) In other words, in the gravitational field of intensity g a beam of light is deflected by angle co, which is proportional to the gravitational field intensity g, to the time t required by the light to traverse the gravitational field, and inversely proportional to the velocity of light c. Or, in general, in a varying gravitational field, like that of the sun, the angle of deflection of a beam of light is : '^^- (18) c /■ F. THE ORBIT OF THE BEAM OF LIGHT We thus see that in a gravitational field a beam of light obeys the same laws as a material body. That is, in the gravitational field of a big mass like that of the sun a beam of light moves in the same kind of orbit as a comet or planet, the only difference in the shape of the orbit being that due to the velocity. Thus, let (in Fig. 17) *S be the sun. At a distance from the sun S a body Pi revolves at a certain velocity. At a certain value Vi of this velocity (about 20 miles per second at 100,000,000 miles distance) this body describes a circle (1), as the planets do approximately. If the velocity is greater, the orbit becomes elongated, taking the form of an ellipse (2), the more so the higher the velocity, until at the velocity yiV2 the orbit becomes infinitely elongated, becoming a parabola (3), as approximated by most comets. That is, the body moves further and further away and slows 60 RELATIVITY AND SPACE down until it comes to rest at infinite distance. At still higher velocity the orbit is a hyperbola (4), and the higher the velocity the straighter the hyperbola (5) becomes, until finally, at the extremely high velocity of light, c = Fig. 17. 186,000 miles per second, the hyperbola (6) becomes almost a straight line. Even if the beam of light comes very close to the sun, the angle of its hyperbolic motion is only 1.7 seconds of arc, that is, about one-thousandth of the diam- eter of the sun. It is, however, still observable during an eclipse, by the apparent shifting of stars near the sun, when the glare of that body is cut off. Verification of the calculation has been made under this condition. As stated above, the difference between Einstein's and Newton's laws of gravitation for velocities of the order of those of the planets and comets is so small that it cannot be observed except in a few cases, as in the motion of the planet Mercury. However, with increasing velocity, the difference increases, and it becomes 100 per cent at the velocity of light. That is, the orbit of the beam of light calculated by Newton's theory of gravitation would give only half the angle given by Einstein's theory, so that it is possible to determine by observation whether Einstein's or Newton's theory is correct. Observations during the last solar eclipses have checked with Einstein's theory of gravi- GRAVITATION AND THE GRAVITATIONAL FIELD 61 tation, and therefore with the relativity theory, on which it is based. G. MATHEMATICAL EFFECTS OF GRAVITATIONAL AND CENTRIFUGAL FIELD As has been seen, centrifugal force is the inertial effect of rotary motion, and the laws of gravitational force are identical with the inertial effect of radial acceleration. We have seen, however, in the preceding lectures, that relative motion affects the length of a body and the time on the body, shortens the length and slows down the time. There- fore, in the centrifugal field as well as in the gravitational field, mathematical effects due to the change of length by the relative motion, and physical effects due to the slowing down of time, may be expected. Suppose we measure a flywheel, first while it is standing still. We measure its diameter and find it d, and then measure the circumference with the same measure and find it C, and we find that the circumference is it times the diameter : C = ird. (19) Now we again measure the flywheel, but while it is rapidly revolving. We stand outside and watch it spinning around while it is being measured. We find the same meas- ure for the diameter d, because the motion of the flywheel is tangential — that is, at right angles to the direction of the measure as it is used in measuring the diameter^ — and the motion thus does not affect the length of the measure. But when I measure the circumference of the revolving flywheel, the motion is in the same direction in which I use the measuring rod, and the length of the measuring rod thus appears shorter to me outside of the flywheel. In other words, the measuring rod is shortened and therefore is con- tained in the circumference a larger number of times than before; that is, for the circumference C we get a larger number than before. Hence: 62 RELATIVITY AND SPACE C>7rd. (20) As the shortening of the rod is by the factor ^/ i _ ^ > it thus is, in the centrifugal field : p _ ird Now let us do the same thing in a gravitational field, that is, a radially accelerated system. We find the circumfer- ence C to be the same as if no gravitation existed, as the acceleration is radial and thus does not affect the measuring rod used in a tangential direction in measuring the circum- ference. When measuring the diameter, however, the measuring rod is shortened, and the diameter thus comes out larger than it would in the absence of a gravitational field. As the circumference C has remained the same and the diameter d has increased in measure, the circumference is not ird any more, but less; that is, in a gravitational field, it is Cx <^ \ No metric re- Triangle, sum of angles .... = 180° < 180° > 180° / lations. Parallels to line through point 1 2(00) 0 i No conception Size of line, plane and space. Infinite and Infinite and Finite in size. of infinity and unlimited. unlimited. but unlim- ited, that is, without boundary. therefore none of parallels. Our space thus varies in character between Euclidean space (1), at great distance from masses, to elliptical space (3), where it is filled by masses, and the average character GRAVITATION AND THE GRAVITATIONAL FIELD 65 of the space thus is eUiptic in correspondence with the average mass density in space. In a centrifugal field space would be hyperbolic. How- ever, where a centrifugal field exists a gravitational field must simultaneously exist, more intense than the centrif- ugal force, to hold the revolving mass toward the center;^ otherwise the mass would go out tangentially and no centrif- ugal force would exist. The effect of the gravitational field thus must always be greater than that of the centrif- ugal field, and the resultant effect is thus an elliptic form of space. Space, that is, our universe, then must be finite, and any straight line, indefinitely extended, would finally run back into itself, close, after a length equal to several hundred million light-years. (A light-year being the distance traveled by light in one year, and light traveling 186,000 miles in one second, a light-year is about six millions of millions of miles). ^ The total volume of the universe, then, would be equal to about 4 X 10^^ cubic miles. 1 Except in a small scale, as a flywheel, where molecular forces, as the cohesion of the material, may counteract the centrifugal force and so keep the revolving mass together. However, then the field intensities are so low and the centrifugal field is so limited that its hyperbolic nature can have no effect on the universe as a whole. 2 The "world radius" (see Lecture IV) is given by Einstein as R^ = 2/Kp, where p is the average density of the mass throughout the universe, and 2/K = 1.08 X 20" cm. Assuming the average distance between the fixed stars as 40 light- years, their average diameter as 1,000,000 miles, and their average density equal to that of water, or 1, the average mass density of the universe would be about: P = 3 X 10-". Thus R = 1026 cm. = 60 X 1020 miles = 100,000,000 light-years. The length of the straight line then would be: I = 4R = 400,000,000 light-years, and the volume of the universe would be V = 2w^R'^ = 4 X 10^3 cubic miles. 5 66 RELATIVITY AND SPACE The expression 4 X 10*^^ does not look so formidable, but let us try to get a conception of it- — for instance, as cents. How long would it take to count 4 X 10^^ cents in money? To expedite the process we may count not in cents, nor in dollars, nor hundred-dollar bills, but in checks, as a check can be made out for a larger amount of money than any bill. We can count out about two checks per second. Let us then make these checks as large as imaginable^ — make each out for the total wealth of the earth^ — that is, the total value of all cities and villages, all fields, forests, mines and factories, all ships and rail- roads, in short everything existing on earth, hundreds of thousands of millions of dollars. Suppose we count out two checks per second, each for the total wealth of the earth, and count out such checks continuously, 24 hours per day, weekdays, Sundays and holidays, and get all the thousand millions of human beings on earth to help us count out such checks, and do that from their birth to their death without ever stopping, and assume that hundreds of thousands of years ago, when man developed from his apelike ancestors, he had, been put to work to count such checks, and throughout all its existence on earth the human race had spent every second to count checks, each for the total wealth of the earth, then, the total amount of money counted out, compared with 4 X 10^^ cents, would not be so large as an acorn is compared with the total earth. Thus it is impossible to get a conception of 4 X 10'^^; to the human mind it is infinite. But, while inconceivable, still it is finite, and one of the conclusions of the relativity theory is that the universe is not infinite, but is finite in volume, though unlimited. The size of the universe is the smaller the larger the mass contained in it. It thus would follow that if the universe were entirely filled with mass, say of the density of water, it would have only a rather limited size — a few hundred million miles diameter. This puts a limit on the size of GRAVITATION AND THE GRAVITATIONAL FIELD 67 masses which can exist in the universe. Our sun, though a miUion times larger than the earth and nearly a million miles in diameter, is one of the smaller fixed stars. Betel- geuse (a Orionis) is estimated to have a diameter of more than 300,000,000 miles — so large that if the sun were placed in its center. Mercury, Venus and the earth, and even Mars, would still be inside of it. But if Betelgeuse had the density of water, it would about fill the whole universe; that is, the universe would have shrunk to the size of Betelgeuse. Or, in other words, the eUiptic char- acter of the universe would be so great that its total volume would only be about as large as Betelgeuse.^ However, Betelgeuse is one of the innumerable stars in a universe so large that the enormous size of Betelgeuse from our earth appears as a mere point without any diam- eter. From this, then, it follows that the density of Betelgeuse must be very low, rather more like a thin gas than a solid. I. TIME EFFECTS In a gravitational field length is shortened, thus giving the changes of the laws of mathematics discussed above. Moreover, time is slowed down, as we have seen in dis- cussing the effect of relative motion. That is, if we bring an accurate clock to the sun- — or better still to one of the giant stars like Betelgeuse^ — when watching it from the earth, we would see it going slower. Now, this experiment can be made and offers the possibility of a further check on the relativity theory. We cannot carry a clock from the earth to Betelgeuse, but we do not need to do this, since every incandescent hydrogen atom, for instance, is an accurate clock, vibrating at a rate definitely fixed by the 1 If the density of the body is p = 1, and this body fills the entire universe, then the world's radius would be: R^ = 1.08 X 10" cm. R = S& X 10>2 cm. = 225,000,000 miles. 68 RELATIVITY AND SPACE electrical constants of the hydrogen atom and showing us the exact rate of its vibration in the spectroscope by the wave length or frequency of its spectrum lines. Thus in a strong gravitational field the frequency of luminous vibrations of the atoms should be found slowed down; in other words, the spectrum lines should be shifted towards the red end of the spectrum. The amount of this shift is so small that it has not yet been possible to prove its existence beyond doubt, but there seems to be some evidence of it.