The velocity of the object in frame O' isAccording to the Galilean transformation, the velocity in frame O isIfthentwice the speed of light. We know in fact that nothing can travel faster than the speed of light, so the Galilean transformation is wrong. Special relativity predicts that velocities be added using the formula

Using this withgives

The frames above are separating, but we can use that velocity is a vector to deal with the case of approaching inertial frames.

Ifandthen

]]>{jatex options:inline}t{/jatex} is the time interval for the observer left on Earth

{jatex options:inline}t_0{/jatex} is the time interval for the astronaut.

{jatex options:inline}t_0=t \sqrt{1-v^2/c^2}=9.711 \times \sqrt{1-0.9^2} =4.23 {/jatex} years to two decimal places.]]>

Any light passing near the hole is made to follow a curve path by the intense curvature of spacetime. Some such are shown below.

As the black hole is approached the gravitational forces and the tidal forces – the difference between the gravitational force of your head and your feet – both increase, causing you to be 'spaghettified'. At some point you will pass through the photon sphere – a thin shell of photons orbiting the hole. As the distance from the black hole decreases, the force of gravity increases and the escape velocity increases. At some distance from the singularity, the escape velocity will equal the speed of light. At this distance, the kinetic energy of a body able to escape will be equal in magnitude to the gravitational potential energy.

The radius above is called the Scharzschild radius. It produces a shell around the hole called the event horizon. The event horizon forms the surface of the hole. Anything passing inside the event horizon cannot escape the hole, and is doomed to meet the singularity.

A black hole the mass of the Sun has a Schwarzschild radius of

]]>3. Time dilation.

3. Bending of light in a gravitational field. Verified by Eddington in 1915 during a solar eclipse.

4. Time dilation in the presence of a gravitational field. , confirmed in the 1970s.

5. Gravitational waves, verified in 2016.

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Anyone carrying a light source would emit light that traveled faster in their direction of travel, and slower to a stationary observer behind the.

Repeated experiments showed however, that the speed of light was a fixed number {jatex options:inline}c=299792458{/jatex}m/s, whatever speed of source and observer. Light, and in fact all electromagnetic radiation did not obey the Galilean transformation.

In addition there was the puzzling form of the Lorentz Force {jatex options:inline}\mathbf{F}=q \mathbf{E}+q \mathbf{v} \times \mathbf{B}{/jatex}.

The Lorentz law implied a force that depended on speed. Because the speed changes under the Galilean Transformation, so should the force, then using {jatex options:inline}\mathbf{F}=m \mathbf{a}{/jatex}, so should the acceleration. This directly contradicts the Galilean transformation, which implies forces and accelerations are unchanged by the Galilean Transformation.]]>

The gravitational mass is the mass that gives rise to gravitation forces via the equation {jatex options:inline}F=\frac{Gm_1m_2}{r^2}{/jatex}.

The inertial mass is the mass a body has because of its resistance to force and acceleration.=br /> All the calculations imply that gravitational and inertial mass have the same value, and this is an assumption of the general theory of relativity. It is important to realise that there is no theoretical basis for the gravitational mass and inertial mass being the same, but so much of theoretical physics depends on it, and there is no evidence to contradict it.]]>

A more extreme effect is gravitational lensing. A massive galaxy can deflect the light from distant sources of light e.g. quasars to produce multiple images of an object. The Massive galaxy is acting as a lens to bring light from the quasar to a focus.

P >]]>

General Relativity tells us that spacetime is a unified four dimensional structure. The presence of matter causes spacetime to become warped. Bodies moving in this warped spacetime appear to be subject to a force because they do not travel in straight lines, but they are actually moving along special curves in spacetime called geodesics. They appear to be subject to a force, but in fact they are moving in warped spacetime. Picture for example, the warping of spacetime by a large mass – say a planet, as below right (the image shows warping in two dimensions, though of course, spacetime is actually four dimensional).

As a particle moves in the region of the mass, it will follow a curved path, being deflected towards the mass/ If the mass increases, the curvature of spacetime becomes more pronounced, and the amount of deflection increases. The warping on a two dimensional sheet can be used to illustrate the orbit of the Earth around the Sun.

The Sun warps space time and the Earth follows a geodesic, a path of shortest possible length in four dimensional spacetime, which in this case is a closed path.

]]>This is not in fact so, and makes the concept of 'frames of reference' so important. A frame of reference is simply an environment in which measurements are made. We can all think of ourselves being at rest in our own frame of reference, and as any frame of reference as being a three dimensional coordinate system, on which we can measure positions of events and distances, along with a clock on which we can record time and time intervals. There is a special class of frames of reference called inertial frames of reference. These are frames of reference with constant velocities, so that the frame itself is not accelerating. Between any two inertial frames of reference, we can to a good approximation use the Galilean Transformation to find coordinates, times in one inertial frame for any event E, and distances and times too, if we have these values in the other.]]>

Space and time are absolute. The time coordinates given to observations in all inertial frames are related by a constant time shift. In particular, all observers in all inertial frames agree on the time interval between any two events.

The physical laws take the same form in all inertial frames. This is the principle of relativity.

Together these two principles imply the Galilean transformation can be used to transform positions, velocities and accelerations between inertial frames. The Galilean transformation implies the acceleration in one inertial frame is the same as the acceleration in any other. Because position and velocity are not the same in all inertial frames, the principle of relativity implies that no physical law can be a function of position or velocity (though a physical law may be a function of distance between two points, since distance does not change between inertial frames).

The discovery of the Lorentz force law for a charged particle moving in a combination of electric and magnetic fields –- does depend explicitly on the velocity. This implies that either this is not a fundamental force of nature, or that Newtonian mechanics is flawed.

In fact the second of these turned out to be the case. The postulates of Newtonian mechanics above were replaced by the postulates of special relativity:

The speed of light is the same in all inertial frames

The physical laws take the same form in all inertial frames.

The first of these is a consequence of Maxwells' Laws. The second is the principle of relativity, which is retained. Time is no longer absolute. Different observers may disagree over the time interval between events and even about the order in which events occur. Physical laws may depend explicitly on velocity (and the Lorentz force law becomes a fundamental law of nature).

The constancy of the speed of light was implied by experiment well before the special theory of relativity was formulated, and was long a source of controversy. Scientists hypothesised a fluid which filled all space and called it the ether. They thought the speed of light could be measured relative to the ether and might have speedcalculated from physical constants, but relative to an observer might have some other speed. In fact they found the speed of light to have the same speed relative to all observers. This is now just one of many experimental pieces of evidence in favour of special relativity.

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